The unphysicality of Hilbert spaces

• URL: http://arxiv.org/abs/2308.06669v2
• Date: Tue, 5 Sep 2023 12:26:52 GMT
• Title: The unphysicality of Hilbert spaces
• Authors: Gabriele Carcassi, Francisco Calderon, Christine A. Aidala
• Abstract summary: We show that spaces should not be considered the correct' spaces to represent quantum states mathematically. We first prove that the requirements posited by complex inner product spaces are physically justified. We then show that completeness in the infinite-dimensional case requires the inclusion of states with infinite expectations.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We show that Hilbert spaces should not be considered the ``correct'' spaces to represent quantum states mathematically. We first prove that the requirements posited by complex inner product spaces are physically justified. We then show that completeness in the infinite-dimensional case requires the inclusion of states with infinite expectations, coordinate transformations that take finite expectations to infinite ones and vice-versa, and time evolutions that transform finite expectations to infinite ones in finite time. This makes Hilbert spaces physically unsound as they model a potential infinity as an actual infinity. We suspect that at least some problems in quantum theory related to infinities may be ultimately caused by the wrong space being used. We strongly believe a better solution can be found, and we look at Schwartz spaces for inspiration, as, among other things, they guarantee that the expectation of all polynomials of position and momentum are finite, their elements are uniquely identified by these expectations, and they are the only space closed under Fourier transform.

Related papers

• Towards entropic uncertainty relations for non-regular Hilbert spaces [44.99833362998488]
  The Entropic Uncertainty Relations (EUR) result from inequalities that are intrinsic to the Hilbert space and its dual with no direct connection to the Canonical Commutation Relations. The analysis of these EUR in the context of singular Hilbert spaces has not been addressed.
  arXiv  Detail & Related papers  (2025-03-24T23:41:50Z)
• Comment on "The unphysicality of Hilbert spaces" (arXiv:2308.06669v3) [0.0]
  This is a thoughtful dissection of the mathematical structure of quantum mechanics. It seeks to pinpoint difficulties inherent in postulating that the physical states are elements of a Hilbert space.
  arXiv  Detail & Related papers  (2025-01-06T10:55:32Z)
• Hilbert space separability and the Einstein-Podolsky-Rosen state [0.0]
  We propose a test that witnesses the non-separability of the Hilbert space. We consider the original Einstein-Podolsky-Rosen (EPR) state as a candidate for possessing nonlocal correlations stronger than any state in a separable Hilbert space.
  arXiv  Detail & Related papers  (2024-12-02T19:00:03Z)
• Is Planckian discreteness observable in cosmology? [47.03992469282679]
  A Planck scale inflationary era produces the scale invariant spectrum of inhomogeneities with very small tensor-to-scalar ratio of perturbations. Here we evoke the possibility that some of the major puzzles in cosmology would have an explanation rooted in quantum gravity.
  arXiv  Detail & Related papers  (2024-05-21T06:53:37Z)
• A non-hermitean momentum operator for the particle in a box [49.1574468325115]
  We show how to construct the corresponding hermitean Hamiltonian for the infinite as well as concrete example. The resulting Hilbert space can be decomposed into a physical and unphysical subspace.
  arXiv  Detail & Related papers  (2024-03-20T12:51:58Z)
• Completely Discretized, Finite Quantum Mechanics [0.0]
  I propose a version of quantum mechanics featuring a discrete and finite number of states that is plausibly a model of the real world. The model is based on standard unitary quantum theory of a closed system with a finite-dimensional Hilbert space.
  arXiv  Detail & Related papers  (2023-07-21T22:11:59Z)
• Measurement phase transitions in the no-click limit as quantum phase transitions of a non-hermitean vacuum [77.34726150561087]
  We study phase transitions occurring in the stationary state of the dynamics of integrable many-body non-Hermitian Hamiltonians. We observe that the entanglement phase transitions occurring in the stationary state have the same nature as that occurring in the vacuum of the non-hermitian Hamiltonian.
  arXiv  Detail & Related papers  (2023-01-18T09:26:02Z)
• Continuous percolation in a Hilbert space for a large system of qubits [58.720142291102135]
  The percolation transition is defined through the appearance of the infinite cluster. We show that the exponentially increasing dimensionality of the Hilbert space makes its covering by finite-size hyperspheres inefficient. Our approach to the percolation transition in compact metric spaces may prove useful for its rigorous treatment in other contexts.
  arXiv  Detail & Related papers  (2022-10-15T13:53:21Z)
• Contextual unification of classical and quantum physics [0.0]
  We develop the idea that the usual formalism of quantum mechanics stops working when countable infinities of particles are encountered. This is because the dimension of the corresponding Hilbert space becomes uncountably infinite, leading to the loss of unitary equivalence. We show that it provides a natural way to describe the "Heisenberg cut", as well as a unified mathematical model including both quantum and classical physics.
  arXiv  Detail & Related papers  (2022-09-03T16:51:19Z)
• Optimal bounds on the speed of subspace evolution [77.34726150561087]
  In contrast to the basic Mandelstam-Tamm inequality, we are concerned with a subspace subject to the Schroedinger evolution. By using the concept of maximal angle between subspaces we derive optimal bounds on the speed of such a subspace evolution. These bounds may be viewed as further generalizations of the Mandelstam-Tamm inequality.
  arXiv  Detail & Related papers  (2021-11-10T13:32:15Z)
• Gapped Quantum Systems: From Higher Dimensional Lieb-Schultz-Mattis to the Quantum Hall Effect [0.0]
  We consider many-body quantum systems on a finite lattice, where the Hilbert space is the tensor product of finite-dimensional Hilbert spaces associated with each site. We are interested in proving uniform bounds on various properties as the size of the lattice tends to infinity.
  arXiv  Detail & Related papers  (2021-11-02T19:25:03Z)
• Why and whence the Hilbert space in quantum theory? [0.0]
  We explain why and how the Hilbert space comes about in quantum theory. An issue of deriving the norm topology may no have a short-length solution but is likely solvable in the affirmative.
  arXiv  Detail & Related papers  (2021-10-07T13:55:47Z)
• Universality of a truncated sigma-model [0.0]
  Even when regularized using a finite lattice, Bosonic quantum field theories possess an infinite dimensional Hilbert space. A qubitization of the $1+1$ dimensionalally free $sigma$-model based on ideas of non-commutative geometry was previously proposed. We provide evidence that it reproduces the physics of the $sigma$-model both in the infrared and the ultraviolet regimes.
  arXiv  Detail & Related papers  (2021-09-15T18:06:07Z)
• Generalized Probabilistic Theories in a New Light [0.0]
  A new answer to the question of why our universe is quantum mechanical rather than classical will be presented. This paper shows that there is still a possibility that there might be a deterministic level from which our universe emerges.
  arXiv  Detail & Related papers  (2021-03-08T21:28:19Z)
• Quantum speed limits for time evolution of a system subspace [77.34726150561087]
  In the present work, we are concerned not with a single state but with a whole (possibly infinite-dimensional) subspace of the system states that are subject to the Schroedinger evolution. We derive an optimal estimate on the speed of such a subspace evolution that may be viewed as a natural generalization of the Fleming bound.
  arXiv  Detail & Related papers  (2020-11-05T12:13:18Z)
• Discrete Hilbert Space, the Born Rule, and Quantum Gravity [0.21320960069210473]
  Quantum gravitational effects suggest a minimal length, or spacetime interval, of order the Planck length. This in turn suggests that Hilbert space itself may be discrete rather than continuous. We discuss this in the context of quantum gravity, showing that discrete models indeed suggest a discrete Hilbert space with minimum norm.
  arXiv  Detail & Related papers  (2020-07-25T14:13:04Z)
• Quantum Geometric Confinement and Dynamical Transmission in Grushin Cylinder [68.8204255655161]
  We classify the self-adjoint realisations of the Laplace-Beltrami operator minimally defined on an infinite cylinder. We retrieve those distinguished extensions previously identified in the recent literature, namely the most confining and the most transmitting.
  arXiv  Detail & Related papers  (2020-03-16T11:37:23Z)

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.