Universal $T$-matrices for quantum Poincaré groups: contractions and quantum reference frames

• URL: http://arxiv.org/abs/2604.01058v1
• Date: Wed, 01 Apr 2026 16:00:18 GMT
• Title: Universal $T$-matrices for quantum Poincaré groups: contractions and quantum reference frames
• Abstract summary: Universal $T$-matrices, or Hopf algebra dual forms, for quantum groups are revisited.<n>A new quantum deformation of the (1+1) centrally extended Poincaré Lie algebra is obtained.<n>The Poincaré Hopf algebra dual form stands as a natural candidate for describing the symmetry structure of relativistic quantum reference frame transformations.
• Score: 0.2519906683279153
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Universal $T$-matrices, or Hopf algebra dual forms, for quantum groups are revisited, and their contraction theory is developed. As a first illustrative example, the (1+1) timelike $κ$-Poincaré $T$-matrix is explicitly worked out. Afterwards, motivated by recent results on the role of the Hopf algebra dual form of a quantum (1+1) centrally extended Galilei group as the algebraic object underlying non-relativistic quantum reference frame transformations, a new quantum deformation of the (1+1) centrally extended Poincaré Lie algebra is obtained, and its universal $T$-matrix is presented. Finally, the Hopf algebra dual form contraction is applied to this Poincaré $T$-matrix, showing that its corresponding non-relativistic counterpart is precisely the Galilei $T$-matrix associated with quantum reference frames. In this way, the Poincaré Hopf algebra dual form introduced here stands as a natural candidate for describing the symmetry structure of relativistic quantum reference frame transformations. In the appropriate basis, the associated quantum Poincaré group is recognized, remarkably, as a non-trivial central extension of the (1+1) spacelike $κ$-Poincaré dual Hopf algebra.

Related papers

• Quantum cellular automata and invertible phases of matter [0.0]
  We study invertible quasi-local algebras over uniformly locally finite metric spaces $X$ with infinite-dimensional local von Neumann algebras.<n>We show that the group of Brauer equivalence classes of such algebras is isomorphic to both the group of phases of invertible states and the group of stable equivalence classes of quantum cellular automata.
  arXiv  Detail & Related papers  (2026-08-26T23:15:46Z)
• Non-Invertible Symmetries Mixing with Witt Non-Trivial Quantum Cellular Automata [0.0]
  Self-dualities and the stacking of symmetry-protected topological phases are basic operations on quantum many-body systems.<n>We realize this entire structure microscopically as quantum cellular automata acting on a certain local operator algebra.
  arXiv  Detail & Related papers  (2026-07-23T18:00:00Z)
• Renormalization flows for 1D mixed states and a quantum Goursat lemma [0.0]
  We study the boundary states of non-chiral two-dimensional topologically ordered models.<n>Data consist of renormalization fixed points built from representations of finite-dimensional $C*$-Hopf algebras.<n>We formulate and prove a quantum generalization of Goursat's lemma for finite-dimensional $C*$-Hopf algebras.
  arXiv  Detail & Related papers  (2026-07-09T14:56:47Z)
• Quantum two-dimensional superintegrable systems in flat space: exact-solvability, hidden algebra, polynomial algebra of integrals [41.99844472131922]
  It includes the Smorodinsky-Winternitz potentials I-II (the Holt potential), the Fokas-Lagerstrom model, the 3-body Calogero and Wolfes (equivalently, $G$ rational, or $I_6$) models.<n>It is shown that all of them are exactly-solvable, thus, confirming the Montreal conjecture.<n>Each model is characterized by infinitely-many finite-dimensional invariant subspaces, which form the infinite flag.
  arXiv  Detail & Related papers  (2025-12-30T07:39:35Z)
• Representation theory of Gaussian unitary transformations for bosonic and fermionic systems [0.0]
  We analyze the behavior of the sign ambiguity that one needs to deal with when moving between the groups of the symplectic and special annihilation group. We show how we can efficiently describe group multiplications in the double cover without the need of going to a faithful representation on an exponentially large or even infinite-dimensional space.
  arXiv  Detail & Related papers  (2024-09-18T01:22:38Z)
• Volichenko-type metasymmetry of braided Majorana qubits [0.0]
  This paper presents different mathematical structures connected with the parastatistics of braided Majorana qubits. Mixed-bracket Heisenberg-Lie algebras are introduced.
  arXiv  Detail & Related papers  (2024-06-02T21:50:29Z)
• Generalized cluster states from Hopf algebras: non-invertible symmetry and Hopf tensor network representation [0.0]
  Cluster states are crucial resources for measurement-based quantum computation (MBQC) We present the construction of cluster states based on Hopf algebras.
  arXiv  Detail & Related papers  (2024-05-15T11:46:52Z)
• Geometry of degenerate quantum states, configurations of $m$-planes and invariants on complex Grassmannians [55.2480439325792]
  We show how to reduce the geometry of degenerate states to the non-abelian connection $A$. We find independent invariants associated with each triple of subspaces. Some of them generalize the Berry-Pancharatnam phase, and some do not have analogues for 1-dimensional subspaces.
  arXiv  Detail & Related papers  (2024-04-04T06:39:28Z)
• Gelfand-Tsetlin basis for partially transposed permutations, with applications to quantum information [0.9208007322096533]
  We study representation theory of the partially transposed permutation matrix algebra. We show how to simplify semidefinite optimization problems over unitary-equivariant quantum channels. We derive an efficient quantum circuit for implementing the optimal port-based quantum teleportation protocol.
  arXiv  Detail & Related papers  (2023-10-03T17:55:10Z)
• Quantum Current and Holographic Categorical Symmetry [62.07387569558919]
  A quantum current is defined as symmetric operators that can transport symmetry charges over an arbitrary long distance. The condition for quantum currents to be superconducting is also specified, which corresponds to condensation of anyons in one higher dimension.
  arXiv  Detail & Related papers  (2023-05-22T11:00:25Z)
• Quantum teleportation in the commuting operator framework [63.69764116066747]
  We present unbiased teleportation schemes for relative commutants $N'cap M$ of a large class of finite-index inclusions $Nsubseteq M$ of tracial von Neumann algebras. We show that any tight teleportation scheme for $N$ necessarily arises from an orthonormal unitary Pimsner-Popa basis of $M_n(mathbbC)$ over $N'$.
  arXiv  Detail & Related papers  (2022-08-02T00:20:46Z)
• Quantum double aspects of surface code models [77.34726150561087]
  We revisit the Kitaev model for fault tolerant quantum computing on a square lattice with underlying quantum double $D(G)$ symmetry. We show how our constructions generalise to $D(H)$ models based on a finite-dimensional Hopf algebra $H$.
  arXiv  Detail & Related papers  (2021-06-25T17:03:38Z)
• A refinement of Reznick's Positivstellensatz with applications to quantum information theory [72.8349503901712]
  In Hilbert's 17th problem Artin showed that any positive definite in several variables can be written as the quotient of two sums of squares. Reznick showed that the denominator in Artin's result can always be chosen as an $N$-th power of the squared norm of the variables.
  arXiv  Detail & Related papers  (2019-09-04T11:46:26Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.