Semiheaps and Ternary Algebras in Quantum Mechanics Revisited

• URL: http://arxiv.org/abs/2111.13369v1
• Date: Fri, 26 Nov 2021 09:09:58 GMT
• Title: Semiheaps and Ternary Algebras in Quantum Mechanics Revisited
• Authors: Andrew James Bruce
• Abstract summary: We re-examine the appearance of semiheaps and (para-associative) ternary algebras in quantum mechanics. New aspect of this work is a discussion of how symmetries of a quantum system induce homomorphisms of the relevant semiheaps and ternary algebras.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We re-examine the appearance of semiheaps and (para-associative) ternary algebras in quantum mechanics. In particular, we review the construction of a semiheap on a Hilbert space and the set of bounded operators on a Hilbert space. The new aspect of this work is a discussion of how symmetries of a quantum system induce homomorphisms of the relevant semiheaps and ternary algebras.

Related papers

• Symmetry-restricted quantum circuits are still well-behaved [45.89137831674385]
  We show that quantum circuits restricted by a symmetry inherit the properties of the whole special unitary group $SU(2n)$. It extends prior work on symmetric states to the operators and shows that the operator space follows the same structure as the state space.
  arXiv  Detail & Related papers  (2024-02-26T06:23:39Z)
• Beyond trace class -- Tensor products of Hilbert spaces and operator ideals in quantum physics [0.0]
  Banach operator ideals are lurking in the foundations and philosophy of quantum physics and quantum information theory. We establish a canonical isomorphism between the Hilbert spaces $Hotimes (K otimes L)$ and $(H otimes K) otimes L$ (Theorem 3.8) and revisit the role of trace class operators. A few applications are specified, including the appropriateness of the class of Hilbert-Schmidt operators and an implied Banach operator ideal representation of the tensor product of two complex spaces $H otimes K$ (Proposition 3.4) and
  arXiv  Detail & Related papers  (2023-08-08T23:37:02Z)
• Trace class operators and states in p-adic quantum mechanics [0.0]
  We show that one can define a suitable space of trace class operators in the non-Archimedean setting. The analogies, but also the several (highly non-trivial) differences, with respect to the case of standard quantum mechanics in a complex Hilbert space are analyzed.
  arXiv  Detail & Related papers  (2022-10-04T12:44:22Z)
• 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)
• Self-adjoint extension schemes and modern applications to quantum Hamiltonians [55.2480439325792]
  monograph contains revised and enlarged materials from previous lecture notes of undergraduate and graduate courses and seminars delivered by both authors over the last years on a subject that is central both in abstract operator theory and in applications to quantum mechanics. A number of models are discussed, which are receiving today new or renewed interest in mathematical physics, in particular from the point of view of realising certain operators of interests self-adjointly.
  arXiv  Detail & Related papers  (2022-01-25T09:45:16Z)
• Reformulation of Quantum Theory [0.0]
  The standard quantum mechanics over a complex Hilbert space, is a Hamiltonian mechanics, regarding the Hilbert space as a linear real manifold equipped with its canonical symplectic form and restricting only to the expectation-value functions of Hermitian operators. We reformulate the structure of quantum mechanics in the language of symplectic manifold and avoid linear structure of Hilbert space in such a way that the results can be stated for an arbitrary symplectic manifold.
  arXiv  Detail & Related papers  (2022-01-03T17:15:35Z)
• Hilbert Space Fragmentation and Commutant Algebras [0.0]
  We study the phenomenon of Hilbert space fragmentation in isolated Hamiltonian and Floquet quantum systems. We use the language of commutant algebras, the algebra of all operators that commute with each term of the Hamiltonian or each gate of the circuit.
  arXiv  Detail & Related papers  (2021-08-23T18:00:01Z)
• 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)
• Hilbert-space geometry of random-matrix eigenstates [55.41644538483948]
  We discuss the Hilbert-space geometry of eigenstates of parameter-dependent random-matrix ensembles. Our results give the exact joint distribution function of the Fubini-Study metric and the Berry curvature. We compare our results to numerical simulations of random-matrix ensembles as well as electrons in a random magnetic field.
  arXiv  Detail & Related papers  (2020-11-06T19:00:07Z)
• Quantum Space, Quantum Time, and Relativistic Quantum Mechanics [0.0]
  We treat space and time as quantum degrees of freedom on an equal footing in Hilbert space. Motivated by considerations in quantum gravity, we focus on a paradigm dealing with linear, first-order translations Hamiltonian and momentum constraints.
  arXiv  Detail & Related papers  (2020-04-20T09:04:15Z)
• 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.