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

Angel Ballesteros, Diego Fernandez-Silvestre, Ivan Gutierrez-Sagredo

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.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.

math.QACross-listedarXiv:2604.01058Apr 2, 2026

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Fedir T. Vasko, O. É. Raichev

Springer eBooksJan 1, 200574 citations

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Rudolf Haag

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Functional Integral Approach to $C^*$ -algebraic Quantum Mechanics I: Heisenberg and PoincaréFunctional Integral Approach to $C^*$-algebraic Quantum Mechanics I: Heisenberg and Poincaré

J. LaChapelle

The algebraic approach to quantum mechanics has been vital to the development of quantum theory since its inception, and it has evolved into a mathematically rigorous $C^\ast$ -algebraic formulation of the theory's axioms. Conversely, the functional approach in the form of Feynman path integrals is far from mathematically rigorous: Nevertheless, path integrals provide an equally valid and useful formulation of the axioms of quantum mechanics. The two approaches can be merged by employing a notion of functional integration based on topological groups that allows to construct functional integral representations of $C^\ast$ -algebras. The merger achieves a hybrid formulation of the axioms of quantum mechanics in which topological groups play a leading role. To illustrate the formalism, we apply the framework to non-relativistic and relativistic quantum mechanics via the Heisenberg and Poincaré groups.The algebraic approach to quantum mechanics has been vital to the development of quantum theory since its inception, and it has evolved into a mathematically rigorous $C^\ast$-algebraic formulation of the theory's axioms. Conversely, the functional approach in the form of Feynman path integrals is far from mathematically rigorous: Nevertheless, path integrals provide an equally valid and useful formulation of the axioms of quantum mechanics. The two approaches can be merged by employing a notion of functional integration based on topological groups that allows to construct functional integral representations of $C^\ast$-algebras. The merger achieves a hybrid formulation of the axioms of quantum mechanics in which topological groups play a leading role. To illustrate the formalism, we apply the framework to non-relativistic and relativistic quantum mechanics via the Heisenberg and Poincaré groups.

arXiv:1505.08102May 27, 20150 citations

Quantum Hamiltonians and Kinetic Equations

Victor Valentinovich Vedenyapin, Alexander V. Sinitsyn, Eugene Dulov

Elsevier eBooksJan 1, 20110 citations

Universal

T

-matrices for quantum Poincaré groups: contractions and quantum reference frames

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.

Functional Integral Approach to

C^*

-algebraic Quantum Mechanics I: Heisenberg and Poincaré

The algebraic approach to quantum mechanics has been vital to the development of quantum theory since its inception, and it has evolved into a mathematically rigorous

C^\ast

-algebraic formulation of the theory's axioms. Conversely, the functional approach in the form of Feynman path integrals is far from mathematically rigorous: Nevertheless, path integrals provide an equally valid and useful formulation of the axioms of quantum mechanics. The two approaches can be merged by employing a notion of functional integration based on topological groups that allows to construct functional integral representations of

C^\ast

-algebras. The merger achieves a hybrid formulation of the axioms of quantum mechanics in which topological groups play a leading role. To illustrate the formalism, we apply the framework to non-relativistic and relativistic quantum mechanics via the Heisenberg and Poincaré groups.