Tagged vector space, Part I: Dirac notation as originally intended

• URL: http://arxiv.org/abs/2510.17327v1
• Date: Mon, 20 Oct 2025 09:21:28 GMT
• Title: Tagged vector space, Part I: Dirac notation as originally intended
• Authors: Filippus S. Roux,
• Abstract summary: A generalization is provided for the notion of tags, as used in various formulations of physical scenarios.<n>It leads to the definition of tagged vector spaces, based on a set of axioms for tags and their extractors.<n>Such a tagged vector space is used to provide, in the context of quantum optics, a formal mathematical description for the Dirac notation.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: A generalization is provided for the notion of tags, as used in various formulations of physical scenarios. It leads to the definition of tagged vector spaces, based on a set of axioms for tags and their extractors. As an application, such a tagged vector space is used to provide, in the context of quantum optics, a formal mathematical description for the Dirac notation that is closer to its intended usage compared to current mathematical formulations: it provides a one-to-one mapping between kets and bras and allows operators to operate either to the left or to the right. The canonical commutation relations for the quadrature and ladder operators are derived as consequences of the axioms of the tagged vector space. These axioms also lead to a symplectic phase space with the Wigner function and the Weyl transform emerging naturally.

Related papers

• A vector logic for intensional formal semantics [0.0]
  Formal semantics and distributional semantics are distinct approaches to linguistic meaning.<n>This paper proves that these frameworks are structurally compatible for intensional semantics.
  arXiv  Detail & Related papers  (2026-02-03T00:24:37Z)
• Symbolic Disentangled Representations for Images [83.88591755871734]
  We propose ArSyD (Architecture for Disentanglement), which represents each generative factor as a vector of the same dimension as the resulting representation.<n>We study ArSyD on the dSprites and CLEVR datasets and provide a comprehensive analysis of the learned symbolic disentangled representations.
  arXiv  Detail & Related papers  (2024-12-25T09:20:13Z)
• PosFormer: Recognizing Complex Handwritten Mathematical Expression with Position Forest Transformer [51.260384040953326]
  Handwritten Mathematical Expression Recognition (HMER) has wide applications in human-machine interaction scenarios. We propose a position forest transformer (PosFormer) for HMER, which jointly optimize two tasks: expression recognition and position recognition. PosFormer consistently outperforms the state-of-the-art methods 2.03%/1.22%/2, 1.83%, and 4.62% gains on datasets.
  arXiv  Detail & Related papers  (2024-07-10T15:42:58Z)
• Transport of Algebraic Structure to Latent Embeddings [8.693845596949892]
  Machine learning often aims to produce latent embeddings of inputs which lie in a larger, abstract mathematical space. How can we learn to "union" two sets using only their latent embeddings while respecting associativity? We propose a general procedure for parameterizing latent space operations that are provably consistent with the laws on the input space.
  arXiv  Detail & Related papers  (2024-05-27T02:24:57Z)
• Subspace Representations for Soft Set Operations and Sentence Similarities [17.52824249186434]
  We realize the representation of word sets and corresponding set operations within pre-trained word embedding spaces. By grounding our approach in the linear subspaces, we enable efficient computation of various set operations. We show that our subspace-based set operations consistently outperform vector-based ones in both sentence similarity and set retrieval tasks.
  arXiv  Detail & Related papers  (2022-10-24T08:34:10Z)
• Unsupervised Disentanglement with Tensor Product Representations on the Torus [78.6315881294899]
  Current methods for learning representations with auto-encoders almost exclusively employ vectors as the latent representations. In this work, we propose to employ a tensor product structure for this purpose. In contrast to the conventional variations methods, which are targeted toward normally distributed features, the latent space in our representation is distributed uniformly over a set of unit circles.
  arXiv  Detail & Related papers  (2022-02-13T04:23:12Z)
• Self-Adjointness of Toeplitz Operators on the Segal-Bargmann Space [62.997667081978825]
  We prove a new criterion that guarantees self-adjointness of Toeplitz operator with unbounded operator-valued symbols. We extend the Berger-Coburn estimate to the case of vector-valued Segal-Bargmann spaces.
  arXiv  Detail & Related papers  (2022-02-09T19:14:13Z)
• Vector Space Semantics for Lambek Calculus with Soft Subexponentials [0.8287206589886879]
  We develop a vector space semantics for Lambek Calculus with Soft Subexponentials. We construct compositional vector interpretations for parasitic gap noun phrases and discourse units with anaphora and ellipsis.
  arXiv  Detail & Related papers  (2021-11-22T16:39:30Z)
• Unsupervised Distillation of Syntactic Information from Contextualized Word Representations [62.230491683411536]
  We tackle the task of unsupervised disentanglement between semantics and structure in neural language representations. To this end, we automatically generate groups of sentences which are structurally similar but semantically different. We demonstrate that our transformation clusters vectors in space by structural properties, rather than by lexical semantics.
  arXiv  Detail & Related papers  (2020-10-11T15:13:18Z)
• Embedding Vector Differences Can Be Aligned With Uncertain Intensional Logic Differences [0.0]
  DeepWalk algorithm is used to assign embedding vectors to nodes in the Atomspace, labeled hypergraph that is used to represent knowledge in the OpenCog AGI system. It is shown that a vector difference operations between embedding vectors are, in appropriate alignable with "intensional difference" operations between the hypergraph nodes corresponding to the embedding vectors.
  arXiv  Detail & Related papers  (2020-05-26T06:20:32Z)
• Word Rotator's Distance [50.67809662270474]
  Key principle in assessing textual similarity is measuring the degree of semantic overlap between two texts by considering the word alignment. We show that the norm of word vectors is a good proxy for word importance, and their angle is a good proxy for word similarity. We propose a method that first decouples word vectors into their norm and direction, and then computes alignment-based similarity.
  arXiv  Detail & Related papers  (2020-04-30T17:48:42Z)

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.