arXiv subjects

Shahn Majid

Publications and source records attributed to Shahn Majid.

At least 55 records · Page 3Linked to original sources

arXiv2020

Quantum geometry of Boolean algebras and de Morgan duality

We take a fresh look at the geometrization of logic using the recently developed tools of `quantum Riemannian geometry' applied in the digital case over the field $\Bbb F_2=\{0,1\}$, extending de Morgan duality to this context of differential forms and connections. The 1-forms correspond to graphs and the exterior derivative of a subset amounts to the arrows that cross between the set and its complement. The line graph $0-1-2$ has a non-flat but Ricci flat quantum Riemannian geometry. The previously known four quantum geometries on the triangle graph, of which one is curved, are revisited in terms of left-invariant differentials, as are the quantum geometries on the dual Hopf algebra, the group algebra of $\Bbb Z_3$. For the square, we find a moduli of four quantum Riemannian geometries, all flat, while for an $n$-gon with $n>4$ we find a unique one, again flat. We also propose an extension of de Morgan duality to general algebras and differentials over $\Bbb F_2$.

math.QA↗

arXiv2020

Quantum Gravity and Riemannian Geometry on the Fuzzy Sphere

We study the quantum geometry of the fuzzy sphere defined as the angular momentum algebra $[x_i,x_j]=2\imathλ_p ε_{ijk}x_k$ modulo setting $\sum_i x_i^2$ to a constant, using a recently introduced 3D rotationally invariant differential structure. Metrics are given by symmetric $3 \times 3$ matrices $g$ and we show that for each metric there is a unique quantum Levi-Civita connection with constant coefficients, with scalar curvature $ \frac{1}{2}({\rm Tr}(g^2)-\frac{1}{2}{\rm Tr}(g)^2)/\det(g)$. As an application, we construct Euclidean quantum gravity on the fuzzy unit sphere. We also calculate the charge 1 monopole for the 3D differential structure.

math.QA↗

arXiv2020

Quantum differentials on cross product Hopf algebras

We construct canonical strongly bicovariant differential graded algebra structures on all four flavours of cross product Hopf algebras, namely double cross products $A\hookrightarrow A\bowtie H\hookleftarrow H$, double cross coproducts $A\twoheadleftarrow A {\blacktriangleright\!\!\blacktriangleleft} H\twoheadrightarrow H$, biproducts $A{\buildrel\hookrightarrow\over \twoheadleftarrow}A{\cdot\kern-.33em\triangleright\!\!\!<} B$ and bicrossproducts $A\hookrightarrow A{\blacktriangleright\!\!\triangleleft} H\twoheadrightarrow H$ on the assumption that the factors have strongly bicovariant calculi $Ω(A),Ω(H)$ (or a braided version $Ω(B)$). We use super versions of each of the constructions. Moreover, the latter three quantum groups all coact canonically on one of their factors and we show that this coaction is differentiable. In the case of the Drinfeld double $D(A,H)=A^{\rm op}\bowtie H$ (where $A$ is dually paired to $H$), we show that its canonical actions on $A,H$ are differentiable. Examples include are a canonical $Ω(\Bbb C_q[GL_2\ltimes \Bbb C^2])$ for the quantum group of affine transformations of the quantum plane and $Ω(\Bbb C_λ[{\rm Poinc_{1,1}}])$ for the bicrossproduct Poincaré quantum group in 2 dimensions. We also show that $Ω(\Bbb C_q[GL_2])$ itself is uniquely determined by differentiability of the canonical coaction on the quantum plane and of the determinant subalgebra.

math.QA↗

arXiv2020

Quantum geometry, logic and probability

Quantum geometry on a discrete set means a directed graph with a weight associated to each arrow defining the quantum metric. However, these `lattice spacing' weights do not have to be independent of the direction of the arrow. We use this greater freedom to give a quantum geometric interpretation of discrete Markov processes with transition probabilities as arrow weights, namely taking the diffusion form $\partial_+ f=(-Δ_θ+ q-p)f$ for the graph Laplacian $Δ_θ$, potential functions $q,p$ built from the probabilities, and finite difference $\partial_+$ in the time direction. Motivated by this new point of view, we introduce a `discrete Schroedinger process' as $\partial_+ψ=\imath(-Δ+V)ψ$ for the Laplacian associated to a bimodule connection such that the discrete evolution is unitary. We solve this explicitly for the 2-state graph, finding a 1-parameter family of such connections and an induced `generalised Markov process' for $f=|ψ|^2$ in which there is an additional source current built from $ψ$. We also discuss our recent work on the quantum geometry of logic in `digital' form over the field $\Bbb F_2=\{0,1\}$, including de Morgan duality and its possible generalisations.

math-ph↗

arXiv2019

Digital finite quantum Riemannian geometries

We study bimodule quantum Riemannian geometries over the field $\Bbb F_2$ of two elements as the extreme case of a finite-field adaptation of noncommutative-geometric methods for physics. We classify all parallelisable such geometries for coordinate algebras up to vector space dimension $n\le 3$, finding a rich moduli of examples for $n=3$ and top form degree 2, including many that are not flat. Their coordinate algebras are commutative but their differentials are not. We also study the quantum Laplacian $Δ=(\ ,\ )\nabla{\rm d}$ on our models and characterise when it has a massive eigenvector.

math.DG↗

arXiv2019

Quantum Riemannian geometry and particle creation on the integer line

We construct noncommutative or `quantum' Riemannian geometry on the integers $\Bbb Z$ as a lattice line $\cdots\bullet_{i-1}-\bullet_i-\bullet_{i+1}\cdots$ with its natural 2-dimensional differential structure and metric given by arbitrary non-zero edge square-lengths $\bullet_i{\buildrel a_i\over -}\bullet_{i+1}$. We find for general metrics a unique $*$-preserving quantum Levi-Civita connection, which is flat if and only if $a_i$ are a geometric progression where the ratios $ρ_i=a_{i+1}/a_i$ are constant. More generally, we compute the Ricci tensor for the natural antisymmetric lift of the volume 2-form and find that the quantum Einstein-Hilbert action up to a total divergence is $-{1\over 2}\sum ρΔρ$ where $(Δρ)_i=ρ_{i+1}+ρ_{i-1}-2ρ_i$ is the standard discrete Laplacian. We take a first look at some issues for quantum gravity on the lattice line. We also examine $1+0$ dimensional scalar quantum theory with mass $m$ and the lattice line as discrete time. As an application, we compute discrete time cosmological particle creation for a step function jump in the metric by a factor $ρ$, finding that an initial vacuum state has at later times an occupancy $ =(1-\sqrtρ)^2/(4\sqrtρ)$ in the continuum limit, independently of the frequency. The continuum limit of the model is the time-dependent harmonic oscillator, now viewed geometrically.

gr-qc↗

arXiv2018

Quantum gravity on a square graph

We consider functional-integral quantisation of the moduli of all quantum metrics defined as square-lengths $a$ on the edges of a Lorentzian square graph. We determine correlation functions and find a fixed relative uncertainty $Δa/ = 1/\sqrt{8}$ for the edge square-lengths relative to their expected value $ $. The expected value of the geometry is a rectangle where parallel edges have the same square-length. We compare with the simpler theory of a quantum scalar field on such a rectangular background. We also look at quantum metric fluctuations relative to a rectangular background, a theory which is finite and interpolates between the scalar theory and the fully fluctuating theory.

gr-qc↗

arXiv2018

Co-double bosonisation and dual bases of c_q[SL_2] and c_q[SL_3]

We find a dual version of a previous double-bosonisation theorem whereby each finite-dimensional braided-Hopf algebra $B$ in the category of comodules of a coquasitriangular Hopf algebra $A$ has an associated coquasitriangular Hopf algebra $B^{\underline{\rm op}}\rtimes A \ltimes B^*$. As an application we find new generators for $c_q[SL_{2}]$ reduced at $q$ a primitive odd root of unity with the remarkable property that their monomials are essentially a dual basis to the standard PBW basis of the reduced Drinfeld-Jimbo quantum enveloping algebra $u_q(sl_{2})$. Our methods apply in principle for general $c_{q}[G]$ as we demonstrate for $c_q[SL_3]$ at certain odd roots of unity.

math.QA↗

arXiv2018

On the emergence of the structure of Physics

We consider Hilbert's problem of the axioms of Physics at a qualitative or conceptual level. This issue is more pressing than ever as we seek to understand how both General Relativity and quantum theory could emerge from some deeper theory of quantum gravity, and in this regard I have previously proposed a `principle of self-duality' or `quantum Born reciprocity' as a key structure. Here I outline some of my recent work around the idea of quantum spacetime as motivated by this non-standard philosophy, including a new toy model of gravity on a spacetime consisting of four points forming a square.

math-ph↗

arXiv2017

Quantum Koszul formula on quantum spacetime

Noncommutative or quantum Riemannian geometry has been proposed as an effective theory for aspects of quantum gravity. Here the metric is an invertible bimodule map $Ω^1\otimes_AΩ^1\to A$ where $A$ is a possibly noncommutative or `quantum' spacetime coordinate algebra and $(Ω^1,d)$ is a specified bimodule of 1-forms or `differential calculus' over it. In this paper we explore the proposal of a `quantum Koszul formula' with initial data a degree -2 bilinear map $\perp$ on the full exterior algebra $Ω$ obeying the 4-term relations \[ (-1)^{|η|} (ωη)\perpζ+(ω\perpη)ζ=ω\perp(ηζ)+(-1)^{|ω|+|η|}ω(η\perpζ),\quad\forallω,η,ζ\inΩ\] and a compatible degree -1 `codifferential' map $δ$. These provide a quantum metric and interior product and a canonical bimodule connection $\nabla$ on all degrees. The theory is also more general than classically in that we do not assume symmetry of the metric nor that $δ$ is obtained from the metric. We solve and interpret the $(δ,\perp)$ data on the bicrossproduct model quantum spacetime $[r,t]=λr$ for its two standard choices of $Ω$. For the $α$-family calculus the construction includes the quantum Levi-Civita connection for a general quantum symmetric metric, while for the more standard $β=1$ calculus we find the quantum Levi-Civita connection for a quantum `metric' that in the classical limit is antisymmetric.

math.QA↗

arXiv2017

Quantum Bianchi identities and characteristic classes via DG categories

We show how DG categories arise naturally in noncommutative differential geometry and use them to derive noncommutative analogues of the Bianchi identities for the curvature of a connection. We also give a derivation of formulae for characteristic classes in noncommutative geometry following Chern's original derivation, rather than using cyclic cohomology. We show that a related DG category for extendable bimodule connections is a monoidal tensor category and in the metric compatible case give an analogue of a classical antisymmetry of the Riemann tensor. The monoidal structure implies the existence of a cup product on noncommutative sheaf cohomology. Another application is to prove that the curvature of a line module reduces to a 2-form on the base algebra. We also extend our geometric approach to Dirac operators. We illustrate the theory on the q-sphere, the permutation group S_3 and the bicrossproduct model quantum spacetime with algebra [r,t]=λr.

math.QA↗

arXiv2017

Smooth dense subalgebras and Fourier multipliers on compact quantum groups

We define and study dense Frechet subalgebras of compact quantum groups consisting of elements rapidly decreasing with respect to an unbounded sequence of real numbers. Further, this sequence can be viewed as the eigenvalues of a Dirac-like operator and we characterize the boundedness of its commutators in terms of the eigenvalues. Grotendieck's theory of topological tensor products immediately yields a Schwartz kernel theorem for linear operators on compact quantum groups and allows us to introduce a natural class of pseudo-differential operators on compact quantum groups. As a by-product, we develop elements of the distribution theory and corresponding Fourier analysis. We give applications of our construction to obtain sufficient conditions for $L^p-L^q$ boundedness of coinvariant linear operators. We provide necessary and sufficient conditions for algebraic differential calculi on Hopf subalgebras of compact quantum groups to extend to the proposed smooth structure. We check explicitly that these conditions hold true on the quantum $SU(2)$ for both its 3-dimensional and 4-dimensional calculi.

math.OA↗

arXiv2017

Classification of digital affine noncommutative geometries

It is known that connected translation invariant $n$-dimensional noncommutative differentials $d x^i$ on the algebra $k[x^1,\cdots,x^n]$ of polynomials in $n$-variables over a field $k$ are classified by commutative algebras $V$ on the vector space spanned by the coordinates. This data also applies to construct differentials on the Heisenberg algebra `spacetime' with relations $[x^μ,x^ν]=λΘ^{μν}$ where $ Θ$ is an antisymmetric matrix as well as to Lie algebras with pre-Lie algebra structures. We specialise the general theory to the field $k={\ \mathbb{F}}_2$ of two elements, in which case translation invariant metrics (i.e. with constant coefficients) are equivalent to making $V$ a Frobenius algebras. We classify all of these and their quantum Levi-Civita bimodule connections for $n=2,3$, with partial results for $n=4$. For $n=2$ we find 3 inequivalent differential structures admitting 1,2 and 3 invariant metrics respectively. For $n=3$ we find 6 differential structures admitting $0,1,2,3,4,7$ invariant metrics respectively. We give some examples for $n=4$ and general $n$. Surprisingly, not all our geometries for $n\ge 2$ have zero quantum Riemann curvature. Quantum gravity is normally seen as a weighted `sum' over all possible metrics but our results are a step towards a deeper approach in which we must also `sum' over differential structures. Over ${\mathbb{F}}_2$ we construct some of our algebras and associated structures by digital gates, opening up the possibility of `digital geometry'.

math.DG↗

arXiv2016

Generalised noncommutative geometry on finite groups and Hopf quivers

We explore the differential geometry of finite sets where the differential structure is given by a quiver rather than as more usual by a graph. In the finite group case we show that the data for such a differential calculus is described by certain Hopf quiver data as familiar in the context of path algebras. We explore a duality between geometry on the function algebra vs geometry on the group algebra, i.e. on the dual Hopf algebra, illustrated by the noncommutative Riemannian geometry of the group algebra of $S_3$. We show how quiver geometries arise naturally in the context of quantum principal bundles. We provide a formulation of bimodule Riemannian geometry for quantum metrics on a quiver, with a fully worked example on 2 points; in the quiver case, metric data assigns matrices not real numbers to the edges of a graph. The paper builds on the general theory in our previous work.

math.QA↗

arXiv2016

Noncommutative spherically symmetric spacetimes at semiclassical order

Working within the recent formalism of Poisson-Riemannian geometry, we completely solve the case of generic spherically symmetric metric and spherically symmetric Poisson-bracket to find a unique answer for the quantum differential calculus, quantum metric and quantum Levi-Civita connection at semiclassical order $O(λ)$. Here $λ$ is the deformation parameter, plausibly the Planck scale. We find that $r,t,dr,dt$ are all forced to be central, i.e. undeformed at order $λ$, while for each value of $r,t$ we are forced to have a fuzzy sphere of radius $r$ with a unique differential calculus which is necessarily nonassociative at order $λ^2$. We give the spherically symmetric quantisation of the FLRW cosmology in detail and also recover a previous analysis for the Schwarzschild black hole, now showing that the quantum Ricci tensor for the latter vanishes at order $λ$. The quantum Laplace-Beltrami operator for spherically symmetric models turns out to be undeformed at order $λ$ while more generally in Poisson-Riemannian geometry we show that it deforms to \[ \square f+{λ\over 2}ω^{αβ}({\rm Ric}^γ_α-S^γ_{;α})(\widehat\nabla_βd f)_γ+ O(λ^2)\] in terms of the classical Levi-Civita connection $\widehat\nabla$, the contorsion tensor $S$, the Poisson-bivector $ω$ and the Ricci curvature of the Poisson-connection that controls the quantum differential structure. The Majid-Ruegg spacetime $[x,t]=λx$ with its standard calculus and unique quantum metric provides an example with nontrivial correction to the Laplacian at order $λ$.

gr-qc↗

arXiv2016

Hodge Star as Braided Fourier Transform

We study super-braided Hopf algebras $Λ$ primitively generated by finite-dimensional right crossed (or Drinfeld-Radford-Yetter) modules $Λ^1$ over a Hopf algebra $A$ which are quotients of the augmentation ideal $A^+$ under right multiplication and the adjoint coaction. Here super-bosonisation $Ω=A\ltimesΛ$ provides a bicovariant differential graded algebra on $A$. We introduce $Λ_{max}$ providing the maximal prolongation, while the canonical braided-exterior algebra $Λ_{min}=B_-(Λ^1)$ provides the Woronowicz exterior calculus. In this context we introduce a Hodge star operator $\sharp$ by super-braided Fourier transform on $B_-(Λ^1)$ and left and right interior products by braided partial derivatives. Our new approach to the Hodge star (a) differs from previous approaches in that it is canonically determined by the differential calculus and (b) differs on key examples, having order 3 in middle degree on $k[S_3]$ with its 3D calculus and obeying the $q$-Hecke relation $\sharp^2=1+(q-q^{-1})\sharp$ in middle degree on $k_q[SL_2]$ with its 4D calculus. Our work also provided a Hodge map on quantum plane calculi and a new starting point for calculi on coquasitriangular Hopf algebras $A$ whereby any subcoalgebra $L\subseteq A$ defines a sub braided-Lie algebra and $Λ^1\subseteq L^*$ provides the required data $A^+\to Λ^1$.

math.QA↗

arXiv2015

Spectral triples from bimodule connections and Chern connections

We give a geometrical construction of Connes spectral triples or noncommutative Dirac operators $D$ starting with a bimodule connection on the proposed spinor bundle. The theory is applied to the example of $M_2(\Bbb C)$, and also applies to the standard $q$-sphere and the $q$-disk with the right classical limit and all properties holding except for $\mathcal J$ now being a twisted isometry. We also describe a noncommutative Chern construction from holomorphic bundles which in the $q$-sphere case provides the relevant bimodule connection.

math.QA↗

arXiv2015

Noncommutative Differentials on Poisson-Lie groups and pre-Lie algebras

We show that the quantisation of a connected simply-connected Poisson-Lie group admits a left-covariant noncommutative differential structure at lowest deformation order if and only if the dual of its Lie algebra admits a pre-Lie algebra structure. As an example, we find a pre-Lie algebra structure underlying the standard 3D differential structure on $\C_q[SU_2]$. At the noncommutative geometry level we show that the enveloping algebra $U(\cm)$ of a Lie algebra $\cm$, viewed as quantisation of $\cm^*$, admits a connected differential exterior algebra of classical dimension if and only if $\cm$ admits a pre-Lie algebra. We give an example where $\cm$ is solvable and we extend the construction to the quantisation of tangent and cotangent spaces of Poisson-Lie groups by using bicross-sum and bosonization of Lie bialgebras. As an example, we obtain natural 6D left-covariant differential structures on the bicrossproduct $\C[SU_2]\lrbicross U_λ(su_2^*)$.

math.QA↗