arXiv subjects

Shahn Majid

Publications and source records attributed to Shahn Majid.

At least 19 recordsLinked to original sources

arXiv2026

Coquasi-bialgebroids and cocycle twisting

We introduce coquasi-bialgebroids over a noncommutative base algebra. Using Takeuchi's $\times_B$-coalgebra formalism, we require the coproduct to remain an algebra map into the Takeuchi product, while the product is associative only up to an invertible normalized $3$-cocycle. This gives a bialgebroid analogue of coquasi-bialgebras and provides a natural framework for cocycle-twisted bialgebroid constructions. We develop the basic theory and prove a twisting theorem by convolution-invertible $2$-cochains. As a main class of examples, we construct coquasi Connes--Moscovici-type bialgebroids on $B\otimes H\otimes B$, where $H$ is a coquasi-bialgebra measuring an algebra $B$, with twisting data $\gamma:H\otimes H\to B$. We also give finite-group examples arising from a subgroup $G\subseteq X$ and a choice of transversal. Finally, under finite projectivity assumptions, we describe the dual quasi-bialgebroid construction and its relation to Drinfeld-type twisting.

math.QA↗

arXiv2026

Geometric Aharonov-Bohm phase effect around a black hole

In recent work, we have explored the novel possibility that the geodesic of a spacetime density flow in GR could be upgraded to an amplitude $\psi$ with density $|\psi|^2$. We now show how exactly the relevant amplitude and velocity flow equations arise in an eikonal approximation of Stueckelberg's proper-time quantum mechanics or Klein-Gordon flow. We also show how the divergence of the velocity field recovers the Raychaudhuri equations for relativistic fluids. In this setting, we demonstrate an Aharonov-Bohm type effect for the phase of $\psi& in motion approaching a black-hole.

gr-qc↗

arXiv2026

Time-slicing quantum spacetimes

For quantum field theory on curved spacetimes, a critical role is played by their foliation into spacelike time-slices at each value $t$ of a coordinate time, with corresponding metric in ADM form. We provide a general construction for the spacetime quantum Levi-Civita connection when each spatial slice is replaced by a quantum Riemannian geometry. This is then fully solved for a class of spatial algebras including fuzzy spheres and for any time-dependent spatial quantum metric, shift 1-form and lapse function. The result takes a particularly simple form if the spatial metric evolves in time according to a first order ODE which, in the case of a fuzzy sphere, requires the spatial metric to rotate in time according to the value at each $t$ of the shift vector. As an application, our results provide in principle fuzzy versions of most (pseudo)-Riemannian manifolds. We also fully solve the case of rotationally invariant spacetimes with angular directions replaced by a discrete circle, including a new $\Bbb Z_n$-FLRW model.

gr-qc↗

arXiv2026

Algebraic approach to quantum gravity IV: applications

We provide a relatively self-contained introduction to the application of quantum spacetime and quantum Riemannian geometry to theoretical physics. Recent successes include calculation of the vacuum energy of spacetime curvature fluctuations in a single-plaquette model of quantum gravity, derivation of the Kaluza-Klein ansatz as a consequence of quantum spacetime, exactly conserved Noether charges from variational calculus on a lattice, and a new theory of classical and quantum geodesics. The latter leads to a theory of generally covariant quantum mechanics applicable in General Relativity with intriguing first results for the case of a black-hole. We discuss several open problems past and present, and how they might be addressed going forward. New results include a phase transition for Euclidean quantum gravity on a 4-pointed star.

gr-qc↗

arXiv2026

Geodesic flows on a black-hole background

A recent notion of geodesic flows which comes out of noncommutative geometry but which is also novel in the classical case is studied in detail for a Schwarzschild spacetime. In this framework, the geodesic velocity field is an independent concept which then defines the flow of a density $\rho$ on spacetime or possibly that of an amplitude wave function $\psi$ with $\rho = |\psi|^2$. The proper time flow parameter $s$ is generated collectively by the flow of matter. We show carefully how the $\rho$ evolution can be justified as modelling a large number of geodesics interpolated as a local density. Using Kruskal-Szekeres coordinates, we show that there are no issues crossing the horizon. A novel feature is that whereas two colliding Gaussian bumps in density $\rho$ merge into a single bump, two colliding wave function $\psi$ bumps of opposite phase merge into a dipole with a different density $|\psi|^2$ profile, providing a potential test of our wave-function hypothesis. We also revisit the Klein-Gordon flow or pseudo-quantum mechanics around a black-hole and find that previously found black-hole atom states and modes generated at the horizon when an area of disturbance approaches it are also present inside the black-hole in a reflected fashion. We argue that the behaviour of the horizon modes across the horizon as well as discretisation of the atomic spectrum depend on quantum gravity corrections at the horizon.

gr-qc↗

arXiv2025

Quantum variational calculus on a lattice

We solve the long-standing problem of variational calculus on a noncommutative space or spacetime for a significant class of models with trivial jet bundle. Our approach entails a quantum version of the Anderson variational double complex $\Omega(J^\infty)$ and includes Euler-Lagrange equations and a partial Noether's theorem. We show in detail how this works for a free field on a $\Bbb Z^m$ lattice regarded as a discrete noncommutative geometry, obtaining the Klein-Gordon equation for a scalar field, including with a general metric and gauge field background, as the Euler-Lagrange equations of motion for an action. In the case of a flat metric we also obtain an exactly on-shell conserved stress-energy tensor and Noether charges for a scalar field on the lattice and modified energy-momentum relations.

hep-th↗

arXiv2025

Kaluza-Klein ansatz from Lorentzian quantum gravity on the fuzzy sphere

If Kaluza-Klein ideas were correct as an explanation of Yang-Mills and General Relativity on spacetime, the extra fibre geometry would have to be a sphere of constant size of the order of 10 Planck lengths, hence subject to quantum gravity corrections. Conversely, it was shown in previous work that modelling such corrections by noncommutative coordinates indeed forces the Kaluza-Klein cylinder ansatz form of the metric, and we now propose that the remaining restrictions needed come from quantum gravity on the fibre. Working with a fuzzy sphere fibre, we find that the expected value of the metric is indeed spherical and we propose that it can be taken as of constant size due to freedom in the renormalisation of divergences. In this way, we outline a mechanism whereby the observed structure of gravity plus Yang-Mills can emerge at low energies as a consequence of quantum gravity effects.

hep-th↗

arXiv2025

Quantum jet Hopf algebroids by cotwist

We introduce a cotwist construction for Hopf algebroids that also entails cotwisting or `quantisation' of the base and which is dual to a previous twisting construction of P. Xu. Whereas the latter applied the construction to the algebra of differential operators on a classical base $B$, we show that the dual of this is the algebra of sections $J(B)$ of the jet bundle and hence that the latter forms a Hopf algebroid. This is constructed for commutative algebras $B$ in a pro-object setting via quotients of the pair Hopf algebroid $B\otimes B$ and can then be deformed by our cotwist construction to give a possibly noncommutative jet Hopf algebroid over a noncommutative base. We also observe in the commutative case that $J^k(B)$ for jets of order $k$ can be identified with $J^1(B_k)$ where $B_k$ denotes $B$ equipped with a certain noncommutative first order differential calculus.

math.QA↗

arXiv2025

Finite group gauge theory on graphs and gravity-like modes

We study gauge theory with finite group $G$ on a graph $X$ using noncommutative differential geometry and Hopf algebra methods with $G$-valued holonomies replaced by gauge fields valued in a `finite group Lie algebra' subset of the group algebra $\mathbb{C} G$ corresponding to the complete graph differential structure on $G$. We show that this richer theory decomposes as a product over the nontrivial irreducible representations $\rho$ with dimension $d_\rho$ of certain noncommutative $U(d_\rho)$-Yang-Mills theories, which we introduce. The Yang-Mills action recovers the Wilson action for a lattice but now with additional terms. We compute the moduli space $\mathcal{A}^\times / \mathcal{G}$ of regular connections modulo gauge transformations on connected graphs $X$. For $G$ Abelian, this is given as expected by phases associated to fundamental loops but with additional $\mathbb{R}_{>0}$-valued modes on every edge resembling the metric for quantum gravity models on graphs. For nonAbelian $G$, these modes become positive-matrix valued modes. We study the quantum gauge field theory in the Abelian case in a functional integral approach, particularly for $X$ the finite chain $A_{n+1}$, the $n$-gon $\mathbb{Z}_n$ and the single plaquette $\mathbb{Z}_2\times \mathbb{Z}_2$. We show that, in stark contrast to usual lattice gauge theory, the Lorentzian version is well-behaved, and we identify novel boundary vs bulk effects in the case of the finite chain. We also consider gauge fields valued in the finite-group Lie algebra corresponding to a general Cayley graph differential calculus on $G$, where we study an obstruction to closure of gauge transformations.

hep-th↗

arXiv2025

Klein-Gordon flow on FLRW spacetimes

We study a new approach to generally covariant quantum mechanics applied in the case of an FLRW cosmological background. For positive spatial curvature we find a discrete series of solutions of the Klein-Gordon equation that can reasonably be called gravitationally bound `cosmological atom' states. For all cases of curvature, these modes, as well as more conventional atomic spatial modes bound by an external potential, extend to solutions of the Klein-Gordon equations viewed as stationary modes of Klein-Gordon quantum mechanics where wavefunctions are over spacetime and evolution is with respect to an external `geodesic time' parameter $s$. For general nonstationary states with fixed spatial eigenvector, the theory reduces to a novel 1-dimensional quantum system on the time $t$ axis with potential $1/a(t)^2$, where $a(t)$ is the Friedmann expansion factor. Its behaviour, and hence the evolution of spatial states, changes critically when the Hubble constant exceeds $2/3$ of the particle mass, as typically occurs during inflation. We also find washout of the evolution of spatial observables at late times and a backward-traveling reflected mode generated when the value of $H$ transitions to a larger value.

gr-qc↗

arXiv2024

*-Hopf algebroids

We introduce a theory of $*$-structures for bialgebroids and Hopf algebroids over a $*$-algebra, defined in such a way that the relevant category of (co)modules is a bar category. We show that if $H$ is a Hopf $*$-algebra then the action Hopf algebroid $A\# H$ associated to a braided-commutative algebra in the category of $H$-crossed modules is a full $*$-Hopf algebroid and the Ehresmann-Schauenburg Hopf algebroid $\mathcal{L}(P,H)$ associated to a Hopf-Galois extension or quantum group principal bundle $P$ with fibre $H$ forms a $*$-Hopf algebroid pair, when the relevant (co)action respects $*$. We also show that Ghobadi's bialgebroid associated to a $*$-differential structure $(\Omega^{1},\rm d)$ on $A$ forms a $*$-bialgebroid pair and its quotient in the pivotal case a $*$-Hopf algebroid pair when the pivotal structure is compatible with $*$. We show that when $\Omega^1$ is simultaneously free on both sides, Ghobadi's Hopf algebroid is isomorphic to $\mathcal{L}(A\#H,H)$ for a smash product by a certain Hopf algebra $H$.

math.QA↗

arXiv2024

Generally covariant quantum mechanics

We obtain generally covariant operator-valued geodesic equations on a pseudo-Riemannian manifold $M$ as part of the construction of quantum geodesics on the algebra $D(M)$ of differential operators. Geodesic motion arises here as an associativity condition for a certain form of first order differential calculus on this algebra in the presence of curvature. The corresponding Schr\"odinger picture has wave functions on spacetime and proper time evolution by the Klein-Gordon operator, with stationary modes being solutions of the Klein-Gordon equation. As an application, we describe gravatom solutions of the Klein-Gordon equations around a Schwarzschild black hole, i.e. gravitationally bound states which far from the event horizon resemble atomic states with the black hole in the role of the nucleus. The spatial eigenfunctions exhibit probability density banding as for higher orbital modes of an ordinary atom, but of a fractal nature approaching the horizon.

gr-qc↗

arXiv2024

Fermions in the fuzzy sphere Kaluza-Klein model

We consider spinors on the total space of a Kaluza-Klein model with fuzzy sphere fibre and geometrically realised Dirac operator on the product. We show that a single massless spinor on the product appears on spacetime as multiplets of spinors with a particular signature of differing masses and $SU(2)$ Yang-Mills charges. For example, for the reduced fuzzy sphere isomorphic to $M_2(\mathbb{C})$, a massless spinor appears as two $SU(2)$ doublets and an $SU(2)$ quadruplet in mass ratios $1:5/3:7/3$. Although such signatures do not appear to exactly match the Standard Model, the paper provides proof of concept of the approach, which can be applied to other noncommutative fibre algebras.

hep-th↗

arXiv2024

Complex structure on quantum-braided planes

We construct a quantum Dolbeault double complex $\oplus_{p,q}\Omega^{p,q}$ on the quantum plane $\Bbb C_q^2$. This solves the long-standing problem that the standard differential calculus on the quantum plane is not a $*$-calculus, by embedding it as the holomorphic part of a $*$-calculus. We show in general that any Nichols-Woronowicz algebra or braided plane $B_+(V)$, where $V$ is an object in an abelian $\Bbb C$-linear braided bar category of real type is a quantum complex space in this sense with a factorisable Dolbeault double complex. We combine the Chern construction on $\Omega^{1,0}$ in such a Dolbeault complex for an algebra $A$ with its conjugate to construct a canonical metric compatible connection on $\Omega^1$ associated to a class of quantum metrics, and apply this to the quantum plane. We also apply this to finite groups $G$ with Cayley graph generators split into two halves related by inversion, constructing such a Dolbeault complex $\Omega(G)$ in this case, recovering the quantum Levi-Civita connection for any edge-symmetric metric on the integer lattice with $\Omega(\Bbb Z)$ now viewed as a quantum complex structure. We also show how to build natural quantum metrics on $\Omega^{1,0}$ and $\Omega^{0,1}$ separately where the inner product in the case of the quantum plane, in order to descend to $\otimes_A$, is taken with values in an $A$-bimodule.

math.QA↗

arXiv2024

Quadratic algebras and idempotent braided sets

We study the Yang-Baxter algebras $A(K,X,r)$ associated to finite set-theoretic solutions $(X,r)$ of the braid relations. We introduce an equivalent set of quadratic relations $\Re\subseteq G$, where $G$ is the reduced Gr\"obner basis of $(\Re)$. We show that if $(X,r)$ is left-nondegenerate and idempotent then $\Re= G$ and the Yang-Baxter algebra is PBW. We use graphical methods to study the global dimension of PBW algebras in the $n$-generated case and apply this to Yang-Baxter algebras in the left-nondegenerate idempotent case. We study the $d$-Veronese subalgebras for a class of quadratic algebras and use this to show that for $(X,r)$ left-nondegenerate idempotent, the $d$-Veronese subalgebra $A(K,X,r)^{(d)}$ can be identified with $A(K,X,r^{(d)})$, where $(X,r^{(d)})$ are all left-nondegenerate idempotent solutions. We determined the Segre product in the left-nondegenerate idempotent setting. Our results apply to a previously studied class of `permutation idempotent' solutions, where we show that all their Yang-Baxter algebras for a given cardinality of $X$ are isomorphic and are isomorphic to their $d$-Veronese subalgebras. In the linearised setting, we construct the Koszul dual of the Yang-Baxter algebra and the Nichols-Woronowicz algebra in the idempotent case, showing that the latter is quadratic. We also construct noncommutative differentials on some of these quadratic algebras.

math.QA↗

arXiv2024

Quantum geometric Wigner construction for $D(G)$ and braided racks

The quantum double $D(G)=\Bbb C(G)\rtimes \Bbb C G$ of a finite group plays an important role in the Kitaev model for quantum computing, as well as in associated TQFT's, as a kind of Poincar\'e group. We interpret the known construction of its irreps, which are quasiparticles for the model, in a geometric manner strictly analogous to the Wigner construction for the usual Poincar\'e group of $\Bbb R^{1,3}$. Irreps are labelled by pairs $(C, \pi)$, where $C$ is a conjugacy class in the role of a mass-shell, and $\pi$ is a representation of the isotropy group $C_G$ in the role of spin. The geometric picture entails $D^\vee(G)\to \Bbb C(C_G)\blacktriangleright\!\!\!\!< \Bbb C G$ as a quantum homogeneous bundle where the base is $G/C_G$, and $D^\vee(G)\to \Bbb C(G)$ as another homogeneous bundle where the base is the group algebra $\Bbb C G$ as noncommutative spacetime. Analysis of the latter leads to a duality whereby the differential calculus and solutions of the wave equation on $\Bbb C G$ are governed by irreps and conjugacy classes of $G$ respectively, while the same picture on $\Bbb C(G)$ is governed by the reversed data. Quasiparticles as irreps of $D(G)$ also turn out to classify irreducible bicovariant differential structures $\Omega^1_{C, \pi}$ on $D^\vee(G)$ and these in turn correspond to braided-Lie algebras $\mathcal{L}_{C, \pi}$ in the braided category of $G$-crossed modules, which we call `braided racks' and study. We show under mild assumptions that $U(\mathcal{L}_{C,\pi})$ quotients to a braided Hopf algebra $B_{C,\pi}$ related by transmutation to a coquasitriangular Hopf algebra $H_{C,\pi}$.

math.QA↗

arXiv2024

Curvature fluctuations in a baby quantum gravity model

Understanding the microscopic behavior of spacetime is critical for developing a theory of quantum gravity and perhaps solving the cosmological constant problem. In this context, it has been proposed that the quantity of interest is the quantum uncertainty in the Ricci scalar and here we investigate this for a discrete baby quantum gravity model based a single square cell. We find that the averaged Ricci scalar vanishes to leading order but has UV-divergent fluctuations. While this behavior is stable under renormalization, it appears not to be under the introduction of a small cosmological constant.

gr-qc↗

arXiv2024

Quantum Gravity: are we there yet?

The turn of the millennium was a time of optimism about an approach to noncommutative geometry inspired by rich mathematical objects called `quantum groups' and its applications to quantum spacetime. This would model quantum gravity effects as noncommutativity of spacetime coordinates and was arguably going to solve quantum gravity itself. It took a further 20 years from that point to develop a particularly suitable formalism of `quantum Riemannian geometry', but this was largely done and has begun to be used to construct baby quantum gravity models. In this article, we obtain new results for state of the art fuzzy sphere and n-gon models in this approach. We also review what are some elements of quantum gravity that we can already see and what are the critical conceptual and mathematical elements that are still missing to more fully achieve this goal.

gr-qc↗