Operational reconstruction of Feynman rules for quantum amplitudes via composition algebras

• URL: http://arxiv.org/abs/2508.14822v1
• Date: Wed, 20 Aug 2025 16:12:11 GMT
• Title: Operational reconstruction of Feynman rules for quantum amplitudes via composition algebras
• Authors: Jens Köplinger, Michael Habeck, Philip Goyal,
• Abstract summary: We revisit an operational model presented in "Origin of complex quantum amplitudes and Feynman's rules"<n>Our methodology establishes clarity by separating axioms from mathematics, choices from physics, and deductions therefrom.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: This paper revisits an operational model presented in "Origin of complex quantum amplitudes and Feynman's rules", Phys. Rev. A 81 (2010), 022109 (P. Goyal, K. H. Knuth, J. Skilling) as part of the Quantum Reconstruction Program, describing transition amplitudes between measurements. Our methodology establishes clarity by separating axioms from mathematics, choices from physics, and deductions therefrom. We carefully evaluate the original model in a coordinate-independent way without requiring a two-dimensional space a priori. All scalar field and vector space axioms are traced from model axioms and observer choices, including additive and multiplicative units and inverses. Known theorems in math classify allowable amplitude algebras as the real associative composition algebras, namely, the two-dimensional (split-)complex numbers and the four-dimensional (split-)quaternions. Observed probabilities are quadratic in amplitudes, akin to the Born rule in physics. We point out select ramifications of postulated model axioms and ways to rephrase observer questions; and advertise broad utility of our work towards follow-on discovery, whether as a consequence, generalization, or alternative. One seemingly minute generalization is sketched in the outlook, with algebraic consequences at the heart of current open questions in mathematics and physics.

Related papers

• Embezzlement of entanglement, quantum fields, and the classification of von Neumann algebras [41.94295877935867]
  We study the quantum information theoretic task of embezzlement of entanglement in the setting of von Neumann algebras.<n>We quantify the performance of a given resource state by the worst-case error.<n>Our findings have implications for relativistic quantum field theory, where type III algebras naturally appear.
  arXiv  Detail & Related papers  (2024-01-14T14:22:54Z)
• Numerical investigations of the extensive entanglement Hamiltonian in quantum spin ladders [9.617349193925188]
  Entanglement constitutes one of the key concepts in quantum mechanics and serves as an indispensable tool in the understanding of quantum many-body systems. We perform extensive numerical investigations of extensive entanglement properties of coupled quantum spin chains.
  arXiv  Detail & Related papers  (2023-11-03T04:06:20Z)
• Classification of dynamical Lie algebras for translation-invariant 2-local spin systems in one dimension [44.41126861546141]
  We provide a classification of Lie algebras generated by translation-invariant 2-local spin chain Hamiltonians. We consider chains with open and periodic boundary conditions and find 17 unique dynamical Lie algebras. In addition to the closed and open spin chains, we consider systems with a fully connected topology, which may be relevant for quantum machine learning approaches.
  arXiv  Detail & Related papers  (2023-09-11T17:59:41Z)
• 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)
• Should we trade off higher-level mathematics for abstraction to improve student understanding of quantum mechanics? [0.0]
  In work at Georgetown University and with edX, we have been teaching a class focused on an operator-forward viewpoint. It teaches quantum mechanics in a representation-independent fashion and allows for most of the math to be algebraic. It also allows for more advanced concepts relevant for quantum sensing, such as squeezed states, to be introduced in a simpler format than is conventionally done.
  arXiv  Detail & Related papers  (2023-04-28T19:30:19Z)
• General quantum algorithms for Hamiltonian simulation with applications to a non-Abelian lattice gauge theory [44.99833362998488]
  We introduce quantum algorithms that can efficiently simulate certain classes of interactions consisting of correlated changes in multiple quantum numbers. The lattice gauge theory studied is the SU(2) gauge theory in 1+1 dimensions coupled to one flavor of staggered fermions. The algorithms are shown to be applicable to higher-dimensional theories as well as to other Abelian and non-Abelian gauge theories.
  arXiv  Detail & Related papers  (2022-12-28T18:56:25Z)
• 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)
• Towards a Probabilistic Foundation of Relativistic Quantum Theory: The One-Body Born Rule in Curved Spacetime [0.0]
  This work is based on generalizing the quantum-mechanical Born rule for determining particle position probabilities to curved spacetime. A principal motivator for this research has been to overcome internal mathematical problems of quantum field theory. The main contribution of this work to the mathematical physics literature is the development of the Lagrangian picture.
  arXiv  Detail & Related papers  (2020-12-09T18:22:18Z)
• Euclidean formulation of relativistic quantum mechanics of N particles [0.0]
  Euclidean formulation of relativistic quantum mechanics for systems of a finite number of degrees of freedom is discussed. Theory can be formulated entirely in the Euclidean representation without the need for analytic continuation. Explicit formulas for generators of the Poincar'e group for any spin are constructed and shown to be self-adjoint on the Euclidean representation of the Hilbert space.
  arXiv  Detail & Related papers  (2020-10-20T15:27:50Z)
• Getting to the Bottom of Noether's Theorem [0.0]
  We show that Noether's theorem holds whenever we can map observables to generators in such a way that each observable generates a one- parameter group that preserves itself. We show this expresses a relation between quantum and statistical mechanics, closely connected to the principle that "inverse temperature is imaginary time"
  arXiv  Detail & Related papers  (2020-06-26T00:56:17Z)
• Models of zero-range interaction for the bosonic trimer at unitarity [91.3755431537592]
  We present the construction of quantum Hamiltonians for a three-body system consisting of identical bosons mutually coupled by a two-body interaction of zero range. For a large part of the presentation, infinite scattering length will be considered.
  arXiv  Detail & Related papers  (2020-06-03T17:54:43Z)
• 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.