Noncommutative integration, quantum mechanics, Tannaka's theorem for compact groupoids and examples

• URL: http://arxiv.org/abs/2303.11752v1
• Date: Tue, 21 Mar 2023 11:18:20 GMT
• Title: Noncommutative integration, quantum mechanics, Tannaka's theorem for compact groupoids and examples
• Authors: Artur O. Lopes, Marcos Sebastian and Victor Vargas
• Abstract summary: We consider topological groupoids in finite and also in a compact settings. In the initial sections, we introduce definitions of typical observables and we studied them in the context of statistical mechanics and quantum mechanics. We exhibit explicit examples and one of them will be the so-called quantum ratchet. This is related to Schwinger's algebra of selective measurements.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We consider topological groupoids in finite and also in a compact settings. In the initial sections, we introduce definitions of typical observables and we studied them in the context of statistical mechanics and quantum mechanics. We exhibit explicit examples and one of them will be the so-called quantum ratchet. This is related to Schwinger's algebra of selective measurements. Here we consider $\mathcal{G}$-kernels, transverse functions, modular functions, and quasi-invariant measures for Haar systems. Later we present our main result which is a version of Tannaka's theorem for Hausdorff compact groupoids - extending the original proof of T. Tannaka.

Related papers

• A Universal Kinematical Group for Quantum Mechanics [0.0]
  In 1968, Dashen and Sharp obtained a certain singular Lie algebra of local densities and currents from canonical commutation relations in nonrelativistic quantum field theory. The corresponding Lie group is infinite dimensional: the natural semidirect product of an additive group of scalar functions with a group of diffeomorphisms.
  arXiv  Detail & Related papers  (2024-04-28T18:46:24Z)
• Fluctuations, uncertainty relations, and the geometry of quantum state manifolds [0.0]
  The complete quantum metric of a parametrized quantum system has a real part and a symplectic imaginary part. We show that for a mixed quantum-classical system both real and imaginary parts of the quantum metric contribute to the dynamics.
  arXiv  Detail & Related papers  (2023-09-07T10:31:59Z)
• Connecting classical finite exchangeability to quantum theory [69.62715388742298]
  Exchangeability is a fundamental concept in probability theory and statistics. We show how a de Finetti-like representation theorem for finitely exchangeable sequences requires a mathematical representation which is formally equivalent to quantum theory.
  arXiv  Detail & Related papers  (2023-06-06T17:15:19Z)
• Hamiltonians and gauge-invariant Hilbert space for lattice Yang-Mills-like theories with finite gauge group [0.0]
  We show that the electric Hamiltonian admits an interpretation as a certain natural, non-unique Laplacian operator on the finite Abelian or non-Abelian group. We illustrate the use of the gauge-invariant basis to diagonalize a dihedral gauge theory on a small periodic lattice.
  arXiv  Detail & Related papers  (2023-01-28T15:16:33Z)
• A singular Riemannian geometry approach to Deep Neural Networks I. Theoretical foundations [77.86290991564829]
  Deep Neural Networks are widely used for solving complex problems in several scientific areas, such as speech recognition, machine translation, image analysis. We study a particular sequence of maps between manifold, with the last manifold of the sequence equipped with a Riemannian metric. We investigate the theoretical properties of the maps of such sequence, eventually we focus on the case of maps between implementing neural networks of practical interest.
  arXiv  Detail & Related papers  (2021-12-17T11:43:30Z)
• Exact thermal properties of free-fermionic spin chains [68.8204255655161]
  We focus on spin chain models that admit a description in terms of free fermions. Errors stemming from the ubiquitous approximation are identified in the neighborhood of the critical point at low temperatures.
  arXiv  Detail & Related papers  (2021-03-30T13:15:44Z)
• Quantum particle across Grushin singularity [77.34726150561087]
  We study the phenomenon of transmission across the singularity that separates the two half-cylinders. All the local realisations of the free (Laplace-Beltrami) quantum Hamiltonian are examined as non-equivalent protocols of transmission/reflection. This allows to comprehend the distinguished status of the so-called bridging' transmission protocol previously identified in the literature.
  arXiv  Detail & Related papers  (2020-11-27T12:53:23Z)
• Particle on the sphere: group-theoretic quantization in the presence of a magnetic monopole [0.0]
  We consider the problem of quantizing a particle on a 2-sphere. We construct the Hilbert space directly from the symmetry algebra. We show how the Casimir invariants of the algebra determine the bundle topology.
  arXiv  Detail & Related papers  (2020-11-10T04:42:08Z)
• On the complex behaviour of the density in composite quantum systems [62.997667081978825]
  We study how the probability of presence of a particle is distributed between the two parts of a composite fermionic system. We prove that it is a non-perturbative property and we find out a large/small coupling constant duality. Inspired by the proof of KAM theorem, we are able to deal with this problem by introducing a cut-off in energies that eliminates these small denominators.
  arXiv  Detail & Related papers  (2020-04-14T21:41:15Z)
• Schwinger's picture of Quantum Mechanics [0.0]
  We will present what can be called Schwinger's foundational approach to Quantum Mechanics. The basic ingredients of this formulation are the textitselective measurements, whose algebra rules define a mathematical structure called groupoid.
  arXiv  Detail & Related papers  (2020-02-21T14:28:31Z)
• 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.