The Gauge Picture of Quantum Dynamics

Kevin Slagle

Department of Electrical and Computer Engineering, Rice University, Houston, Texas 77005 USA
Department of Physics, California Institute of Technology, Pasadena, California 91125, USA
Institute for Quantum Information and Matter and Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, California 91125, USA

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

Although local Hamiltonians exhibit local time dynamics, this locality is not explicit in the Schrödinger 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.

The two most famous pictures of quantum dynamics are the Schrodinger and Heisenberg pictures. In Schrodinger's picture, the wavefunction evolves in time, while in Heisenberg's picture the wavefunction is constant but the operators evolve in time. In this work, we introduce a new picture of quantum dynamics, the gauge picture, which makes deep connections to locality of information and gauge theory.

Regarding locality: A nice advantage of Heisenberg's picture is that locality is explicit in the equations of motion. That is, the time evolution of a local operator only depends on the state of nearby local operators. In contrast, locality is not explicit in this way in Schrodinger's picture, for which there is a single wavefunction whose time dynamics depends on operators everywhere in space. Our new gauge picture modifies Schrodinger's picture such that we can calculate a "local wavefunction" that carries the same information as Schrodinger's wavefunction, expect the time dynamics of local wavefunctions in the gauge picture only depends on nearby Hamiltonian terms, which makes locality explicit in the equations of motion. In order to achieve this explicit locality, the gauge picture adds gauge fields to the equations of motion.

Gauge theory establishes a deep connection between a Hamiltonian (or Lagrangian) with a global symmetry and another Hamiltonian where the global symmetry is replaced by a local gauge symmetry via the addition dynamical gauge fields. Interestingly, Schrodinger's equation $i\hbar \partial_t |\psi\rangle = H |\psi\rangle$ admits a global unitary invariance given by the transformation $|\psi\rangle \to U |\psi\rangle$ and $H \to UHU^\dagger$. Our work shows that it is also possible to apply gauge theory to this global invariance in Schrodinger's equation to obtain a new equation of motion, i.e. the gauge picture, with dynamical gauge fields and a local gauge invariance.

► BibTeX data

► References

[1] David Deutsch and Patrick Hayden. ``Information flow in entangled quantum systems''. Proceedings of the Royal Society of London Series A 456, 1759 (2000). arXiv:quant-ph/​9906007.
https:/​/​doi.org/​10.1098/​rspa.2000.0585
arXiv:quant-ph/9906007

[2] Michael A. Levin and Xiao-Gang Wen. ``String-net condensation: A physical mechanism for topological phases''. Phys. Rev. B 71, 045110 (2005). arXiv:cond-mat/​0404617.
https:/​/​doi.org/​10.1103/​PhysRevB.71.045110
arXiv:cond-mat/0404617

[3] T. Senthil, Ashvin Vishwanath, Leon Balents, Subir Sachdev, and Matthew P. A. Fisher. ``Deconfined Quantum Critical Points''. Science 303, 1490–1494 (2004). arXiv:cond-mat/​0311326.
https:/​/​doi.org/​10.1126/​science.1091806
arXiv:cond-mat/0311326

[4] Beni Yoshida. ``Exotic topological order in fractal spin liquids''. Phys. Rev. B 88, 125122 (2013). arXiv:1302.6248.
https:/​/​doi.org/​10.1103/​PhysRevB.88.125122
arXiv:1302.6248

[5] Kevin Hartnett. ``Matrix multiplication inches closer to mythic goal''. Quanta Magazine (2021). url: https:/​/​www.quantamagazine.org/​mathematicians-inch-closer-to-matrix-multiplication-goal-20210323/​.
https:/​/​www.quantamagazine.org/​mathematicians-inch-closer-to-matrix-multiplication-goal-20210323/​

[6] Volker Strassen. ``Gaussian elimination is not optimal''. Numerische Mathematik 13, 354–356 (1969).
https:/​/​doi.org/​10.1007/​BF02165411

[7] Kevin Slagle. ``Quantum Gauge Networks: A New Kind of Tensor Network''. Quantum 7, 1113 (2023). arXiv:2210.12151.
https:/​/​doi.org/​10.22331/​q-2023-09-14-1113
arXiv:2210.12151

[8] Román Orús. ``A practical introduction to tensor networks: Matrix product states and projected entangled pair states''. Annals of Physics 349, 117–158 (2014). arXiv:1306.2164.
https:/​/​doi.org/​10.1016/​j.aop.2014.06.013
arXiv:1306.2164

[9] Michael P. Zaletel and Frank Pollmann. ``Isometric Tensor Network States in Two Dimensions''. Phys. Rev. Lett. 124, 037201 (2020). arXiv:1902.05100.
https:/​/​doi.org/​10.1103/​PhysRevLett.124.037201
arXiv:1902.05100

[10] Steven Weinberg. ``Testing quantum mechanics''. Annals of Physics 194, 336–386 (1989).
https:/​/​doi.org/​10.1016/​0003-4916(89)90276-5

[11] N. Gisin. ``Weinberg's non-linear quantum mechanics and supraluminal communications''. Physics Letters A 143, 1–2 (1990).
https:/​/​doi.org/​10.1016/​0375-9601(90)90786-N

[12] Joseph Polchinski. ``Weinberg's nonlinear quantum mechanics and the einstein-podolsky-rosen paradox''. Phys. Rev. Lett. 66, 397–400 (1991).
https:/​/​doi.org/​10.1103/​PhysRevLett.66.397

[13] Kevin Slagle. ``Testing Quantum Mechanics using Noisy Quantum Computers'' (2021). arXiv:2108.02201.
arXiv:2108.02201

[14] Brian Swingle. ``Unscrambling the physics of out-of-time-order correlators''. Nature Physics 14, 988–990 (2018).
https:/​/​doi.org/​10.1038/​s41567-018-0295-5

[15] Ignacio García-Mata, Rodolfo A. Jalabert, and Diego A. Wisniacki. ``Out-of-time-order correlators and quantum chaos'' (2022). arXiv:2209.07965.
arXiv:2209.07965

[16] Rahul Nandkishore and David A. Huse. ``Many-Body Localization and Thermalization in Quantum Statistical Mechanics''. Annual Review of Condensed Matter Physics 6, 15–38 (2015). arXiv:1404.0686.
https:/​/​doi.org/​10.1146/​annurev-conmatphys-031214-014726
arXiv:1404.0686

[17] Dmitry A. Abanin, Ehud Altman, Immanuel Bloch, and Maksym Serbyn. ``Colloquium: Many-body localization, thermalization, and entanglement''. Reviews of Modern Physics 91, 021001 (2019). arXiv:1804.11065.
https:/​/​doi.org/​10.1103/​RevModPhys.91.021001
arXiv:1804.11065

[18] Bruno Nachtergaele and Robert Sims. ``Much Ado About Something: Why Lieb-Robinson bounds are useful'' (2011). arXiv:1102.0835.
arXiv:1102.0835

[19] Daniel A. Roberts and Brian Swingle. ``Lieb-robinson bound and the butterfly effect in quantum field theories''. Phys. Rev. Lett. 117, 091602 (2016). arXiv:1603.09298.
https:/​/​doi.org/​10.1103/​PhysRevLett.117.091602
arXiv:1603.09298

[20] Zhiyuan Wang and Kaden R.A. Hazzard. ``Tightening the lieb-robinson bound in locally interacting systems''. PRX Quantum 1, 010303 (2020). arXiv:1908.03997.
https:/​/​doi.org/​10.1103/​PRXQuantum.1.010303
arXiv:1908.03997

Cited by

[1] Sayak Guha Roy and Kevin Slagle, "Interpolating between the gauge and Schrödinger pictures of quantum dynamics", SciPost Physics Core 6 4, 081 (2023).

[2] Kevin Slagle, "Quantum Gauge Networks: A New Kind of Tensor Network", Quantum 7, 1113 (2023).

The above citations are from SAO/NASA ADS (last updated successfully 2024-06-22 12:19:26). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref's cited-by service no data on citing works was found (last attempt 2024-06-22 12:19:24).