nLab compact QED

Idea

What is called compact QED (following Polyakov 1975, 1977, 1987 §4.3) is quantum electrodynamics (QED) with the compactness of the $U(1)$ gauge group taken properly into account (which is not typically done in perturbation theory).

The pleonastic terminology results from the infamous tradition in parts of theoretical physics to conflate Lie groups — here the compact $U(1)$ or the non-compact $\mathbb{R}$ — , with their Lie algebras, which here in both cases is $\mathfrak{u}(1) \simeq \mathbb{R}$. After this conflation, people have to speak of a “compact $U(1)$” to indicate that they really mean the circle group, cf. Preskill 1984 p. 471.

In particular, in the context of “compact QED” one considers magnetic monopole backgrounds and condensates which in 3 dimensions are argued (Polyakov 1975, 1977, 1997) to exhibit confinement in an abelian version of the dual superconductor model of color confinement.

References

The original discussion:

• Alexander M. Polyakov: Compact Gauge Fields and the Infrared Catastrophe, Physics Letters B 59 1 (1975) [doi:10.1016/0370-2693(75)90162-8]

• Alexander M. Polyakov: Quark confinement and topology of gauge theories, Nuclear Physics B 120 3 (1977) 429–458 [doi:10.1016/0550-3213(77)90086-4]

• Alexander Polyakov, §4.3 in: Gauge Fields and Strings, Routledge, Taylor and Francis (1987, 2021) [doi:10.1201/9780203755082, oapen:20.500.12657/50871]

• Alexander Polyakov: Confining strings, Nuclear Physics B 486 1–2 (1997) 23–33 [doi:10.1016/S0550-3213(96)00601-3]

See also:

• Physics.SE: Compact QED and Non-compact QED — Polyakov textbook [physics.SE:q/202626]

Further discussion:

• J. Jersak et al.: Charge Renormalization in Compact Lattice QED, Nucl. Phys. B 251 (1985) 299–310 [doi:10.1016/0550-3213(85)90263-9]

• Ian I. Kogan, Alex Kovner: Compact $QED_3$ — a simple example of a variational calculation in a gauge theory, Phys. Rev. D 51 (1995) 1948–1955 [doi:10.1103/PhysRevD.51.1948, arXiv:hep-th/9410067]

• D. V. Antonov: Various properties of compact QED and confining strings, Phys. Lett. B 428 (1998) 3–4 [doi:10.1016/S0370-2693(98)00419-5, arXiv:hep-th/9802056]

• Orlando Oliveira, Lee C. Loveridge, Paulo J. Silva: Compact QED: the photon propagator, confinement and positivity violation for the pure gauge theory, EPJ Web Conf. 274 02004 (2022) [doi:10.1051/epjconf/202227402004, arXiv:2211.12593]

• M. Cristina Diamantini, Fernando Quevedo, Carlo A. Trugenberger, Luis Zapata: Universal Confining Strings: From Compact QED to the Hadron Spectrum [arXiv:2605.13791]