I would like to find papers, reviews or books where I can read more information about compact QED in different dimensions. I have read the chapter about compact QED in Polyakov's books but it is so short and there is luck of details, derivations etc.

My google skills were useless: I don't find more information about this topic.


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  • $\begingroup$ Possible duplicates: physics.stackexchange.com/q/274338/2451 and links therein. $\endgroup$ – Qmechanic Mar 15 at 8:37
  • $\begingroup$ @Qmechanic I would like to emphasize that I am interested in specific QED modification with compact abelian group. It is not the duplicate, in my view. $\endgroup$ – Artem Alexandrov Mar 15 at 8:41
  • $\begingroup$ Standard QED is with a compact abelian gauge group. $\endgroup$ – Qmechanic Mar 15 at 8:42
  • $\begingroup$ @Qmechanic may be I was incorrect. I mean the QED with 4-potential which can change on the finite interval and periodic. $\endgroup$ – Artem Alexandrov Mar 15 at 8:44
  • $\begingroup$ Are you talking about lattice gauge theory? $\endgroup$ – Qmechanic Mar 15 at 12:21

Polyakov's analysis of compact QED (without fermions) is reviewed in detail in section 8.1 in the book

The book is well-written and engaging. The link given above includes free previews of each chapter, so you can get a feeling for the author's style. The book assumes familiarity with quantum field theory (QFT), specifically quantum chromodynamics (QCD). Lattice gauge theory is the starting point for most of the analyses. The focus is on understanding why confinement occurs in QCD and related models, often citing results of numerical (lattice) calculations as evidence for or against various ideas.

Compact QED is analyzed in section 8.1 as a mathematically-easier model with a confinement phase. This section can probably be read on its own, referring to previous chapters mainly for motivation and maybe a few notational conventions. Here are a couple of excerpts from section 8.1 ("Magnetic monopoles in compact QED") to confirm its relevance:

Compact QED in three and four dimensions has monopole excitations, and these excitations are responsible for the confinement of electric charge. The confinement property exists only at strong lattice couplings in $D=4$ dimensions, but it is found at all lattice couplings in $D=3$ dimensions. The word `compact' refers to the compactness of the $U(1)$ gauge group in the lattice (as opposed to the continuum) formulation of electrodynamics.

Polyakov's demonstration of confinement in compact QED$_3$ is quite beautiful... so the calculation is worth displaying... in a little more detail.

$D=3$ and $D=4$ are the only cases considered in section 8.1, but section 2.4 ("Possible phases of a gauge theory") mentions this about $D=2$:

In $D = 2$ dimensions it is easy to demonstrate that the only phase that exists is the magnetically disordered phase, for any lattice action of the form (2.24) [sum over plaquettes], and for any gauge group. Take the $Z_2$ gauge theory for simplicity... [details follow]

[Conclusion:] The underlying reason for magnetic disorder in $D = 2$ dimensions is the absence of a Bianchi constraint relating different components of the field strength tensor.

Compact QED is sometimes analyzed indirectly by regarding the $U(1)$ gauge group as a limiting case ($N\to\infty$) of the discrete $\mathbb{Z}_N$ gauge group, whose elements are $\exp(2\pi i n/N)$. The preceding excerpt refers to the case $N=2$.

Here's a summary of some of the context leading up to chapter 8 ("Monopoles, calorons, and dual superconductivity"): Chapter 3 goes into some depth comparing various possible definitions of "confinement" and explains why the author chooses to focus on a definition based on magnetic disorder. Section 3.4 introduces the idea of center symmetry, and chapter 4 concludes that center symmetry is closely associated with confinement in QCD. Compact QED is studied in chapter 8, where confinement mechanisms in various simpler models are compared and contrasted with the confinement mechanism in QCD. A major theme in chapter 8 is that the "abelian" confinement mechanism that operates in some simpler models may differ in important ways from the confinement mechanism in QCD.

  • $\begingroup$ It is the most complete answer which I have ever seen! Thank you, Dan! $\endgroup$ – Artem Alexandrov Mar 15 at 17:32
  • $\begingroup$ Dan, for $D=4$ the action for monopoles will have the similar structure but with 4D propagator, doesn't it? $\endgroup$ – Artem Alexandrov Mar 15 at 21:52
  • $\begingroup$ @ArtemAlexandrov Since I haven't worked through it myself recently, I'll refer you to a paper that compares the $D=4$ derivation to the $D=3$ derivation: Banks and Myerson (1977), "Phase transitions in abelian lattice gauge theories" (link to pdf: hep.physics.uoc.gr/Erasmus-IP-2013/files/lectures/Rabinovici/…). Section 2 derives the monopole-gas action for $D=3$, and section 4 summarizes the differences in the case $D=4$. Greensite's book might compare them, too, but I don't have access to the book right now, so I can't check. $\endgroup$ – Chiral Anomaly Mar 16 at 18:39

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