The Gauge Picture of Quantum Dynamics

• URL: http://arxiv.org/abs/2210.09314v5
• Date: Fri, 15 Mar 2024 01:58:58 GMT
• Title: The Gauge Picture of Quantum Dynamics
• Authors: Kevin Slagle,
• Abstract summary: Locality is not explicit in the Schr"odinger picture in the sense that the wavefunction amplitudes do not obey a local equation of motion. We show that locality can be achieved explicitly in the equations of motion by "gauging" the global unitary invariance of quantum mechanics into a local gauge invariance.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Although local Hamiltonians exhibit local time dynamics, this locality is not explicit in the Schr\"{o}dinger picture in the sense that the wavefunction amplitudes do not obey a local equation of motion. We show that geometric locality can be achieved explicitly in the equations of motion by "gauging" the global unitary invariance of quantum mechanics into a local gauge invariance. That is, expectation values $\langle \psi|A|\psi \rangle$ are invariant under a global unitary transformation acting on the wavefunction $|\psi\rangle \to U |\psi\rangle$ and operators $A \to U A U^\dagger$, and we show that it is possible to gauge this global invariance into a local gauge invariance. To do this, we replace the wavefunction with a collection of local wavefunctions $|\psi_J\rangle$, one for each patch of space $J$. The collection of spatial patches is chosen to cover the space; e.g. we could choose the patches to be single qubits or nearest-neighbor sites on a lattice. Local wavefunctions associated with neighboring pairs of spatial patches $I$ and $J$ are related to each other by dynamical unitary transformations $U_{IJ}$. The local wavefunctions are local in the sense that their dynamics are local. That is, the equations of motion for the local wavefunctions $|\psi_J\rangle$ and connections $U_{IJ}$ are explicitly local in space and only depend on nearby Hamiltonian terms. (The local wavefunctions are many-body wavefunctions and have the same Hilbert space dimension as the usual wavefunction.) We call this picture of quantum dynamics the gauge picture since it exhibits a local gauge invariance. The local dynamics of a single spatial patch is related to the interaction picture, where the interaction Hamiltonian consists of only nearby Hamiltonian terms. We can also generalize the explicit locality to include locality in local charge and energy densities.

Related papers

• Response theory for locally gapped systems [0.0]
  We introduce a notion of a emphlocal gap for interacting many-body quantum lattice systems. We prove the validity of response theory and Kubo's formula for localized perturbations in such settings.
  arXiv  Detail & Related papers  (2024-10-14T17:59:29Z)
• Interpolating Between the Gauge and Schr\"odinger Pictures of Quantum Dynamics [0.0]
  spatial locality is not explicit in the Schr"odinger picture equations of motion. The gauge picture is a modification of Schr"odinger's picture such that locality is explicit in the equations of motion.
  arXiv  Detail & Related papers  (2023-07-05T15:33:20Z)
• Double-scale theory [77.34726150561087]
  We present a new interpretation of quantum mechanics, called the double-scale theory. It is based on the simultaneous existence of two wave functions in the laboratory reference frame. The external wave function corresponds to a field that pilots the center-of-mass of the quantum system. The internal wave function corresponds to the interpretation proposed by Edwin Schr"odinger.
  arXiv  Detail & Related papers  (2023-05-29T14:28:31Z)
• Entanglement and localization in long-range quadratic Lindbladians [49.1574468325115]
  Signatures of localization have been observed in condensed matter and cold atomic systems. We propose a model of one-dimensional chain of non-interacting, spinless fermions coupled to a local ensemble of baths. We show that the steady state of the system undergoes a localization entanglement phase transition by tuning $p$ which remains stable in the presence of coherent hopping.
  arXiv  Detail & Related papers  (2023-03-13T12:45:25Z)
• Unitarily inequivalent local and global Fourier transforms in multipartite quantum systems [0.0]
  Local Fourier transforms in each subsystem are defined and related phase space methods are discussed. A global Fourier transform is then defined and related phase space methods are discussed. Time evolution of the system in terms of both local and global variables is discussed.
  arXiv  Detail & Related papers  (2023-01-28T09:16:22Z)
• Localization in the random XXZ quantum spin chain [55.2480439325792]
  We study the many-body localization (MBL) properties of the Heisenberg XXZ spin-$frac12$ chain in a random magnetic field. We prove that the system exhibits localization in any given energy interval at the bottom of the spectrum in a nontrivial region of the parameter space.
  arXiv  Detail & Related papers  (2022-10-26T17:25:13Z)
• Prethermalization and the local robustness of gapped systems [0.22843885788439797]
  We prove that prethermalization is a generic property of gapped local many-body quantum systems. We infer the robustness of quantum simulation in low-energy subspaces.
  arXiv  Detail & Related papers  (2022-09-22T18:00:01Z)
• Dynamics, symmetries, anomaly and vortices in a rotating cosmic string background [0.0]
  Non-relativistic conformally invariant systems in a rotating cosmic string (conical) spacetime are analyzed. Solutions of the equations of motion are found by employing a local canonical transformation. The cone parameter $alpha$ and the angular velocity $Omega$ of the background determine the existence of hidden symmetries.
  arXiv  Detail & Related papers  (2021-09-11T02:22:37Z)
• Anharmonic oscillator: a solution [77.34726150561087]
  The dynamics in $x$-space and in $(gx)-space corresponds to the same energy spectrum with effective coupling constant $hbar g2$. A 2-classical generalization leads to a uniform approximation of the wavefunction in $x$-space with unprecedented accuracy.
  arXiv  Detail & Related papers  (2020-11-29T22:13:08Z)
• Perelman's Ricci Flow in Topological Quantum Gravity [62.997667081978825]
  In our quantum gravity, Perelman's $tau$ turns out to play the role of a dilaton for anisotropic scale transformations. We show how Perelman's $cal F$ and $cal W$ entropy functionals are related to our superpotential.
  arXiv  Detail & Related papers  (2020-11-24T06:29:35Z)
• Anisotropy-mediated reentrant localization [62.997667081978825]
  We consider a 2d dipolar system, $d=2$, with the generalized dipole-dipole interaction $sim r-a$, and the power $a$ controlled experimentally in trapped-ion or Rydberg-atom systems. We show that the spatially homogeneous tilt $beta$ of the dipoles giving rise to the anisotropic dipole exchange leads to the non-trivial reentrant localization beyond the locator expansion.
  arXiv  Detail & Related papers  (2020-01-31T19:00:01Z)

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.