Operators, continuum limits, and gravitational dynamics of quantum-mechanical hair in local discrete gauge theories: the open questions of Preskill and Krauss (1990)

When a continuous gauge group is broken to a discrete subgroup H, the classical long-range fields are screened, yet H-charge survives as 'quantum-mechanical hair' detectable at arbitrarily large distances through Aharonov-Bohm phases acquired by strings (vortices) that wind around the charge. Preskill and Krauss (1990) established this at operator level for abelian H and semiclassically for nonabelian H, and explicitly left open: (i) a gauge-invariant operator realization of non-center, nonabelian H-charges in the presence of vortices; (ii) a well-defined H-charge in finite volume with quantum vortex fluctuations; (iii) string-creation operators and an LSZ-type scattering framework with string asymptotic states; (iv) whether a 3+1-dimensional continuum quantum field theory with parametrically light dynamical strings and an enduring local-versus-global discrete distinction can exist; (v) the dynamical role of discrete hair in gravitational collapse and black-hole evaporation; and (vi) anomaly and consistency conditions for discrete gauge theories formulated directly rather than by embedding. The task is to determine whether a 3+1-dimensional continuum quantum field theory with parametrically light dynamical strings and an enduring local-versus-global discrete distinction can exist (item (iv)); whether a genuine LSZ-type scattering formalism for string asymptotic states can be constructed or proven impossible (the LSZ part of item (iii)); whether discrete quantum-mechanical hair carries information out of an evaporating black hole (the information-theoretic part of item (v)); and whether the anomaly and consistency conditions for discrete gauge theories formulated directly rather than by embedding can be completed (item (vi)).

Background

From continuous to discrete gauge symmetry. In a gauge theory, a condensate of a scalar field of charge $N$ can break a continuous gauge group $G$ down to a discrete subgroup $H$; the standard example is $U(1)\to Z_N$, the cyclic group of order $N$, because the condensate is invariant only under gauge rotations by multiples of $2\pi/N$. Krauss and Wilczek (1989) stressed that such a residual local (gauged) discrete symmetry is physically distinct from an ordinary global discrete symmetry: a gauged symmetry is a redundancy of the variables, so its conservation law cannot be evaded by state preparation or by exotic processes, whereas a global symmetry is merely a law of nature that unknown physics could violate. Preskill and Krauss (1990) worked out the observable consequences of this distinction in quantum field theory.

Quantum-mechanical hair. When $U(1)$ breaks to $Z_N$ the gauge boson acquires a mass and the classical electric field of a $Z_N$-charged particle is exponentially screened, so no classical Gauss-law flux measurement at infinity can read the charge. However, the broken phase contains topologically stable line defects — cosmic strings in three spatial dimensions, vortices in two — carrying magnetic flux $\Phi=2\pi/(Ne)$, where $e$ is the gauge coupling. An object of discrete charge $q$ transported around such a string acquires the Aharonov-Bohm phase $e^{2\pi iq/N}$, measurable in interference experiments at arbitrarily long range even though no classical field escapes. Preskill and Krauss realized this at operator level through the flux operator $F(\Sigma)=\exp[(2\pi i/Ne)\int_\Sigma \mathbf{E}\cdot d\mathbf{s}]$ on a large closed surface $\Sigma$ — equivalently the insertion of a string world sheet in the Euclidean path integral — with the limiting prescription $\lim F(\Sigma)/\langle F(\Sigma)\rangle = e^{2\pi iQ/N}$, where dividing by the vacuum expectation value removes the area-law suppression caused by quantum fluctuations of light charges near the surface. Discrete gauge charge is thus a superselection label (no gauge-invariant local operator can change it) that is detectable at infinity by purely quantum-mechanical means — quantum hair. Since a black hole can carry any charge measurable at infinity, this implies that black holes can carry discrete gauge hair, notwithstanding the classical no-hair theorems.

Nonabelian discrete groups and Cheshire charge. When the unbroken group $H$ is nonabelian, the magnetic flux of a string, defined by parallel transport around a loop based at a point, is only defined up to conjugation $a\mapsto hah^{-1}$ with $h\in H$; winding one string loop through another conjugates its flux (a holonomy interaction). A constant global $H$ transformation cannot be imposed in the presence of such fluxes unless it commutes with all of them, so only elements of the center of $H$ are globally realizable. Consequently, surface operators of the $F(\Sigma)$ type exist only for central charges, and Preskill and Krauss could characterize charges transforming under non-central irreducible representations of $H$ only indirectly, via nonabelian Aharonov-Bohm scattering. Semiclassically they also showed that a closed loop of string (or a vortex-antivortex pair) can carry $H$-charge with no localized source anywhere — Cheshire charge — which is exchanged with point particles winding through the loop; illustrative models include the Pin(2) Alice string, which flips the sign of electric charge upon circumnavigation, and vortex pairs in a theory with unbroken quaternion group. A complementary modern language is the quantum double $D(H)$ (the Drinfeld double, a Hopf algebra built from $H$ and its dual): Kitaev's exactly solvable lattice model for a discrete gauge theory with group $H$ has gauge-invariant closed string (ribbon) operators whose sectors are labelled by pairs consisting of a conjugacy class of $H$ and an irreducible representation of its centralizer, which is the systematic classification of charge-flux sectors that the 1990 continuum analysis lacked.

The open questions. Preskill and Krauss closed their analysis by flagging what remained unresolved: an operator realization of non-center nonabelian charges and a controlled definition of H-charge in finite volume; explicit gauge-invariant operators that create the strings or vortex pairs participating in nonabelian Aharonov-Bohm scattering, and a Lehmann-Symanzik-Zimmermann (LSZ) type reduction — the standard device that defines S-matrix elements from correlation functions of local operators — generalized to string asymptotic states; the existence of a continuum limit with dynamical, parametrically light strings in 3+1 dimensions; the dynamical consequences of quantum hair in gravitational collapse and black-hole evaporation; and the anomaly conditions governing chiral matter in discrete gauge theories formulated directly, without embedding in a continuous gauge group.

Problem Statement

The 1990 analysis of Preskill and Krauss left a coherent, incompletely executed objective: to give a complete quantum field theory of local discrete gauge symmetries with dynamical strings, in which charge sectors are operator-realizable, string states have a well-defined scattering theory, the continuum and gravitational limits are controlled, and the consistency conditions on matter content are known. Concretely, the unresolved questions flagged in that paper are: (i) construct a concrete gauge-invariant operator, or set of operators, whose eigenvalues unambiguously label the non-center irreducible representations of a nonabelian discrete gauge group $H$ in the presence of vortices or strings, or prove that no such operators exist; (ii) give a rigorous prescription under which the $H$-charge of a region is well defined at finite volume despite quantum fluctuations of virtual vortex pairs that can render the enclosed flux fluctuating, and quantify the convergence of the finite-volume definition to the infinite-volume charge; (iii) construct explicit gauge-invariant operators that create the cosmic string or vortex pair participating in nonabelian Aharonov-Bohm scattering, and establish an LSZ-type reduction relating an S-matrix with string asymptotic states to gauge-invariant correlation functions, or prove that no such reduction can exist; (iv) determine whether a continuum quantum field theory in 3+1 dimensions with dynamical strings of parametrically small tension relative to the ultraviolet cutoff, retaining a sharp physical distinction between local and global discrete symmetry, can exist — in particular whether renormalization of the string tension precludes such a limit; (v) determine the dynamical role of discrete quantum-mechanical hair in gravitational collapse and black-hole evaporation, including whether it carries information out of an evaporating black hole; and (vi) complete the anomaly and consistency conditions on chiral matter content for discrete gauge theories formulated directly as effective theories, without embedding in a continuous gauge group. Items (i), (ii) and the operator part of (iii) have since been resolved, and the semiclassical part of (v) has been carried out. The items still open are item (iv), the LSZ part of item (iii), the information-theoretic part of item (v), and item (vi). An obstruction or no-go argument resolves an item that is posed as an existence question.

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