aiqbench analysis
This paper advances theoretical quantum physics by constructing exact lattice models for non-compact gauge groups like SL(2,R), extending standard Kogut-Susskind formulations to include continuous spectra and infinite-dimensional representations. The work bridges concepts from general relativity and dilaton gravity with integrable quantum many-body systems, offering new analytical insights into the dynamics of fermionic matter coupled to non-compact fields.
Non-compact Kogut-Susskind gauge theories: 1D lattice models with exact ${\rm SL}(2,\mathbb{R})$ gauge invariance
arXiv:2611.06771v1 Announce Type: new Abstract: The Kogut-Susskind (KS) formulation of gauge theories constructs quantum many-body Hamiltonians with gauge invariance under a compact Lie group exactly implemented on a spatial lattice. Here, we revisit this construction for a non-compact gauge group. Although motivated in part by theories like general relativity whose first-order formulation involves the non-compact Lorentz group as a local symmetry, we focus on simpler cases in one spatial dimension akin to the Schwinger model. In a bottom-up approach, we show how exact gauge invariance under the simplest non-compact Lie group ${\rm SL}(2,\mathbb{R})$ can be retained exactly on a 1D lattice, and highlight the differences that arise with respect to compact gauge groups, including a continuous spectrum of the Casimir operators of the gauge fields and the emergence of infinite-dimensional representations of the matter fields. For the pure gauge theory, we discuss how a consistent restriction to the positive semigroup ${\rm SL}^{+}(2,\mathbb{R})$ reproduces well-known results from the quantization of Jackiw-Teitelboim (JT) gravity. Motivated by the constraint structure of general dilaton gravity, we introduce static charges coupled to the non-compact gauge field, which can be realised by single-fermion degrees of freedom living in an additional, synthetic dimension. With static charges our lattice model remains exactly integrable, but when extending to dynamical fermionic matter, we arrive at an interacting 1D ${\rm SL}(2,\mathbb{R})$ KS gauge theory. We give an interpretation of this model as a chain of charges sequentially connected by strings whose tension pulls matter left or right depending on the sign of the relative continuous ${\rm SL}(2,\mathbb{R})$ fluxes. This picture becomes exact when the involved ${\rm SL}(2,\mathbb{R})$ charges are strongly self-interacting, where we solve the reduced two-body problem analytically.
Abstract:The Kogut-Susskind (KS) formulation of gauge theories constructs quantum many-body Hamiltonians with gauge invariance under a compact Lie group exactly implemented on a spatial lattice. Here, we revisit this construction for a non-compact gauge group. Although motivated in part by theories like general relativity whose first-order formulation involves the non-compact Lorentz group as a local symmetry, we focus on simpler cases in one spatial dimension akin to the Schwinger model. In a bottom-up approach, we show how exact gauge invariance under the simplest non-compact Lie group ${\rm SL}(2,\mathbb{R})$ can be retained exactly on a 1D lattice, and highlight the differences that arise with respect to compact gauge groups, including a continuous spectrum of the Casimir operators of the gauge fields and the emergence of infinite-dimensional representations of the matter fields. For the pure gauge theory, we discuss how a consistent restriction to the positive semigroup ${\rm SL}^{+}(2,\mathbb{R})$ reproduces well-known results from the quantization of Jackiw-Teitelboim (JT) gravity. Motivated by the constraint structure of general dilaton gravity, we introduce static charges coupled to the non-compact gauge field, which can be realised by single-fermion degrees of freedom living in an additional, synthetic dimension. With static charges our lattice model remains exactly integrable, but when extending to dynamical fermionic matter, we arrive at an interacting 1D ${\rm SL}(2,\mathbb{R})$ KS gauge theory. We give an interpretation of this model as a chain of charges sequentially connected by strings whose tension pulls matter left or right depending on the sign of the relative continuous ${\rm SL}(2,\mathbb{R})$ fluxes. This picture becomes exact when the involved ${\rm SL}(2,\mathbb{R})$ charges are strongly self-interacting, where we solve the reduced two-body problem analytically.
arXiv:2611.06771 [quant-ph]
(or arXiv:2611.06771v1 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2611.06771 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.
Full text under the publisher's open licence · 364 words