Semidual Kitaev lattice model and tensor network representation

Kitaev’s lattice models are usually defined as representations of the Drinfeld quantum double D(H) = H ⋈ H*op, as an example of a double cross product quantum group. We propose a new version based instead on M(H) = Hcop ⧑ H as an example of Majid’s bicrossproduct quantum group, related by semidualisation or ‘quantum Born reciprocity’ to D(H). Given a finite-dimensional Hopf algebra H, we show that a quadrangulated oriented surface defines a representation of the bicrossproduct quantum group Hcop ⧑ H. Even though the bicrossproduct has a more complicated and entangled coproduct, the construction of this new model is relatively natural as it relies on the use of the covariant Hopf algebra actions. Working locally, we obtain an exactly solvable Hamiltonian for the model and provide a definition of the ground state in terms of a tensor network representation.

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 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 space–time as motivated by this non-standard philosophy, including a new toy model of gravity on a space–time consisting of four points forming a square. This article is part of the theme issue ‘Hilbert’s sixth problem’.

Quantum Mechanics on a Curved Snyder Space

We study the representations of the three-dimensional Euclidean Snyder-de Sitter algebra. This algebra generates the symmetries of a model admitting two fundamental scales (Planck mass and cosmological constant) and is invariant under the Born reciprocity for exchange of positions and momenta. Its representations can be obtained starting from those of the Snyder algebra and exploiting the geometrical properties of the phase space that can be identified with a Grassmannian manifold. Both the position and momentum operators turn out to have a discrete spectrum.

BORN RECIPROCITY IN TWO DIMENSIONS: QUAPLECTIC GROUP SYMMETRY, BRANCHING RULES AND STATE LABELLING

We consider the continuous symmetry group underlying Born's reciprocity principle, namely the so-called quaplectic group, the semidirect product of time-space-energy-momentum coordinate transformations with the Weyl-Heisenberg group. In two dimensional Minkowski space this group is Q(1, 1) ≅ U(1, 1) ⋉ H(2), or in Euclidean space Q(2) ≅ U(2) ⋉ H(2). For the 'scalar' system in the sense of induced representations, unitary irreducible representations are carried on a Fock space equivalent to that used by Schwinger as a model of the SU(2) angular momentum algebra, or by Holman and Biedenharn as a model of SU(1,1). Using this construction we consider the branching rules and state labelling problem for the reduction of Q(2) and Q(1, 1) to the 'physical' Euclidean and Poincaré subalgebras, respectively. The results serve to illustrate the difficulties of any consideration of Born reciprocity as an extended symmetry principle of nature.