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
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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 $|psirangle to U |psirangle$ 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 $|psi_Jrangle$, 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 $|psi_Jrangle$ 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.
Featured image: In Schrodinger’s picture, an initial wavefunction $|psi_0rangle$ evolves to $U(t) |psi_0rangle$ after a time $t$, where $U(t) = e^{-iHt}$ is the unitary time evolution operator. The gauge picture instead considers local wavefunctions $|psi_I(t)rangle$ that are associated with a subset (or patch) $I$ in space. The time-evolved local wavefunction $|psi_I(t)rangle = tilde{U}_I^dagger(t) |psi(t)rangle$ is obtained from Schrodinger’s wavefunction $|psi(t)rangle = U(t) |psi_0rangle$ via the unitary operator $tilde{U}_I^dagger(t)$, which reverses the time evolution outside of the region $I$. As a result, the local wavefunction dynamics $partial_I |psi_I(t)rangle$ only depends on nearby Hamiltonian terms that overlap with the region $I$. The figure depicts these unitary operators as quantum circuits, and demonstrates that much of the time evolution from $U(t)$ cancels out with $tilde{U}_I^dagger(t)$, leaving only an hourglass-shaped time evolution operator acting on the initial wavefunction (on the right of the figure). This hourglass-shaped operator is analogous to the light-cone shaped operator growth in Heisenberg’s picture.
Popular summary
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 $ihbar partial_t |psirangle = H |psirangle$ admits a global unitary invariance given by the transformation $|psirangle to U |psirangle$ 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.
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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 2024-03-21 22:55:07). 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 2024-03-21 22:55:05).
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.
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- Source: https://quantum-journal.org/papers/q-2024-03-21-1295/
