On the Kostant-Souriau prequantization of scalar fields with polysymplectic structures

• URL: http://arxiv.org/abs/2411.04087v1
• Date: Wed, 06 Nov 2024 18:07:31 GMT
• Title: On the Kostant-Souriau prequantization of scalar fields with polysymplectic structures
• Authors: Tom McClain,
• Abstract summary: This approach is based on using the natural geometric structures of polysymplectic Hamiltonian field theory to produce an analog of the Kostant-Souriau prequantization map familiar from geometric quantization. I show that while the resulting operators are quite different from those of canonical quantum field theory, the approach is nonetheless able to reproduce a few of canonical quantum field theory's most fundamental results.
• Score: 0.0
• License:
• Abstract: In this paper, I present a novel, purely differential geometric approach to the quantization of scalar fields, with a special focus on the familiar case of Minkowski spacetimes. This approach is based on using the natural geometric structures of polysymplectic Hamiltonian field theory to produce an analog of the Kostant-Souriau prequantization map familiar from geometric quantization. I show that while the resulting operators are quite different from those of canonical quantum field theory, the approach is nonetheless able to reproduce a few of canonical quantum field theory's most fundamental results. I finish by elaborating the current limitations of this approach and briefly discussing future prospects.

Related papers

• Classical and quantum field theory in a box with moving boundaries: A numerical study of the Dynamical Casimir Effect [0.0]
  We present a detailed description of a quantum scalar field theory within a flat spacetime confined to a cavity with perfectly reflecting moving boundaries. We establish an equivalence between this time-dependent setting and a field theory on an acoustic metric with static Dirichlet boundary conditions.
  arXiv  Detail & Related papers  (2024-04-09T09:43:39Z)
• Poincaré symmetries and representations in pseudo-Hermitian quantum field theory [0.0]
  This paper explores quantum field theories with pseudo-Hermitian Hamiltonians. PT-symmetric Hamiltonians serve as a special case. We extend the Poincar'e algebra to include non-Hermitian generators.
  arXiv  Detail & Related papers  (2023-07-31T16:17:20Z)
• The Quantum Path Kernel: a Generalized Quantum Neural Tangent Kernel for Deep Quantum Machine Learning [52.77024349608834]
  Building a quantum analog of classical deep neural networks represents a fundamental challenge in quantum computing. Key issue is how to address the inherent non-linearity of classical deep learning. We introduce the Quantum Path Kernel, a formulation of quantum machine learning capable of replicating those aspects of deep machine learning.
  arXiv  Detail & Related papers  (2022-12-22T16:06:24Z)
• 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)
• Entanglement dynamics of spins using a few complex trajectories [77.34726150561087]
  We consider two spins initially prepared in a product of coherent states and study their entanglement dynamics. We adopt an approach that allowed the derivation of a semiclassical formula for the linear entropy of the reduced density operator.
  arXiv  Detail & Related papers  (2021-08-13T01:44:24Z)
• Towards a Geometrization of Quantum Complexity and Chaos [0.0]
  We show how the restriction of the Quantum Geometric to manifold of states that can be generated through local interactions provides a new tool to understand the consequences of locality in physics.
  arXiv  Detail & Related papers  (2021-07-14T08:43:07Z)
• Stochastic Quantization of Relativistic Theories [0.0]
  It was shown recently that quantization can be made into a well defined quantization scheme on (pseudo-)Riemannian using second order differential geometry. We show that restrictions to relativistic theories can be obtained from this theory by imposing a energy-momentum relation.
  arXiv  Detail & Related papers  (2021-03-03T16:11:58Z)
• 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)
• State preparation and measurement in a quantum simulation of the O(3) sigma model [65.01359242860215]
  We show that fixed points of the non-linear O(3) sigma model can be reproduced near a quantum phase transition of a spin model with just two qubits per lattice site. We apply Trotter methods to obtain results for the complexity of adiabatic ground state preparation in both the weak-coupling and quantum-critical regimes. We present and analyze a quantum algorithm based on non-unitary randomized simulation methods.
  arXiv  Detail & Related papers  (2020-06-28T23:44:12Z)
• From a quantum theory to a classical one [117.44028458220427]
  We present and discuss a formal approach for describing the quantum to classical crossover. The method was originally introduced by L. Yaffe in 1982 for tackling large-$N$ quantum field theories.
  arXiv  Detail & Related papers  (2020-04-01T09:16:38Z)
• Tautological Tuning of the Kostant-Souriau Quantization Map with Differential Geometric Structures [0.0]
  This paper introduces an alternative approach to coordinate-independent quantization called tautologically tuned quantization. In its focus on physically important functions, tautologically tuned quantization hews much more closely to the ad hoc approach of canonical quantization than either traditional geometric quantization or deformation quantization.
  arXiv  Detail & Related papers  (2020-03-25T16:17:01Z)

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.