3bc7492eafa36717ee538291c024a1bc entropy-27-00250-v2.pdf 5d0f8e6d3091290947efa5f0856e69208c6de175 entropy-27-00250-v2.pdf 813f07628d0b5a5f453f565d3eb6616426139fe18dc62ee9b21410369ac49465 entropy-27-00250-v2.pdf Title: Mean Field Approaches to Lattice Gauge Theories: A Review Subject: Due to their broad applicability, gauge theories (GTs) play a crucial role in various areas of physics, from high-energy physics to condensed matter. Their formulations on lattices, lattice gauge theories (LGTs), can be studied, among many other methods, with tools coming from statistical mechanics lattice models, such as mean field methods, which are often used to provide approximate results. Applying these methods to LGTs requires particular attention due to the intrinsic local nature of gauge symmetry, how it is reflected in the variables used to formulate the theory, and the breaking of gauge invariance when approximations are introduced. This issue has been addressed over the decades in the literature, yielding different conclusions depending on the formulation of the theory under consideration. In this article, we focus on the mean field theoretical approach to the analysis of GTs and LGTs, connecting both older and more recent results that, to the best of our knowledge, have not been compared in a pedagogical manner. After a brief overview of mean field theory in statistical mechanics and many-body systems, we examine its application to pure LGTs with a generic compact gauge group. Finally, we review the existing literature on the subject, discussing the results obtained so far and their dependence on the formulation of the theory. Keywords: lattice gauge theory; mean field theory; gauge-invariant approaches Author: Pierpaolo Fontana and Andrea Trombettoni Creator: LaTeX with hyperref Producer: pdfTeX-1.40.25 CreationDate: Fri Feb 28 09:34:41 2025 CET ModDate: Fri Feb 28 10:43:45 2025 CET Custom Metadata: no Metadata Stream: no Tagged: no UserProperties: no Suspects: no Form: none JavaScript: no Pages: 25 Encrypted: no Page size: 595.276 x 841.89 pts (A4) Page rot: 0 File size: 649458 bytes Optimized: no PDF version: 1.7 name type encoding emb sub uni object ID ------------------------------------ ----------------- ---------------- --- --- --- --------- JJZXBU+URWPalladioL-Roma Type 1 Custom yes yes yes 10 0 RAYVNI+URWPalladioL-Bold Type 1 Custom yes yes yes 16 0 BXMNYP+URWPalladioL-Ital Type 1 Custom yes yes yes 21 0 SOSTRQ+CMR10 Type 1 Builtin yes yes yes 26 0 KXUSVI+CMSY10 Type 1 Builtin yes yes yes 31 0 GJCXGY+PazoMath-Italic Type 1 Builtin yes yes yes 60 0 FOBNGI+CMEX10 Type 1 Builtin yes yes yes 65 0 QBJQEB+PazoMath Type 1 Builtin yes yes yes 70 0 ASZYGD+MSBM10 Type 1 Builtin yes yes yes 78 0 SYFPBV+CMMI10 Type 1 Builtin yes yes yes 83 0 SOSTRQ+CMR10 Type 1 Custom yes yes no 93 0 SYFPBV+CMMI10 Type 1 Custom yes yes no 97 0 WUWDGV+CMMI12 Type 1 Custom yes yes no 101 0 KXUSVI+CMSY10 Type 1 Custom yes yes no 107 0 EEICHW+CMR12 Type 1 Custom yes yes no 111 0 CSHVCL+CMMI9 Type 1 Custom yes yes no 117 0 KXUSVI+CMSY10 Type 1 Custom yes yes no 125 0 SOSTRQ+CMR10 Type 1 Custom yes yes no 128 0 WUWDGV+CMMI12 Type 1 Custom yes yes no 131 0 SYFPBV+CMMI10 Type 1 Custom yes yes no 134 0 CSHVCL+CMMI9 Type 1 Custom yes yes no 137 0 EEICHW+CMR12 Type 1 Custom yes yes no 140 0 BCDEEE+CambriaMath CID TrueType Identity-H yes yes yes 174 0 BCDFEE+Calibri TrueType WinAnsi yes yes no 182 0 SOSTRQ+CMR10 Type 1 Custom yes yes no 212 0 WUWDGV+CMMI12 Type 1 Custom yes yes no 215 0 EEICHW+CMR12 Type 1 Custom yes yes no 218 0 CSHVCL+CMMI9 Type 1 Custom yes yes no 221 0 SOSTRQ+CMR10 Type 1 Custom yes yes no 226 0 SYFPBV+CMMI10 Type 1 Custom yes yes no 229 0 WUWDGV+CMMI12 Type 1 Custom yes yes no 232 0 KXUSVI+CMSY10 Type 1 Custom yes yes no 235 0 CSHVCL+CMMI9 Type 1 Custom yes yes no 238 0 Jhove (Rel. 1.28.0, 2023-05-18) Date: 2025-04-10 02:44:37 CEST RepresentationInformation: entropy-27-00250-v2.pdf ReportingModule: PDF-hul, Rel. 1.12.4 (2023-03-16) LastModified: 2025-04-09 13:23:22 CEST Size: 649458 Format: PDF Version: 1.7 Status: Well-Formed and valid SignatureMatches: PDF-hul MIMEtype: application/pdf PDFMetadata: Objects: 577 FreeObjects: 1 IncrementalUpdates: 0 DocumentCatalog: PageLayout: SinglePage PageMode: UseNone Outlines: Item: Title: Introduction Destination: section.1 Item: Title: Reminder on Mean Field Theory Destination: section.2 Children: Item: Title: An Application to Spin Models Destination: subsection.2.1 Item: Title: An Application to Many-Body Systems: The BCS Theory Destination: subsection.2.2 Item: Title: Gauge Symmetry in Quantum Field Theory Destination: section.3 Children: Item: Title: Equations of Motion and Bianchi Identity Destination: subsection.3.1 Item: Title: Parallel Transport Destination: subsection.3.2 Item: Title: Lattice Gauge Theories Destination: section.4 Children: Item: Title: Lagrangian Formulation Destination: subsection.4.1 Item: Title: Hamiltonian Formulation Destination: subsection.4.2 Children: Item: Title: Bosonic Quantum Link Models Destination: subsubsection.4.2.1 Item: Title: Fermionic Quantum Link Models Destination: subsubsection.4.2.2 Item: Title: Elitzur's Theorem Destination: subsection.4.3 Item: Title: Reformulations in Terms of Gauge Invariants Destination: section.5 Children: Item: Title: Recombinations of Degrees of Freedom Destination: subsection.5.1 Item: Title: Field Strength Reformulations Destination: subsection.5.2 Item: Title: Wilson Loop Reformulations Destination: subsection.5.3 Item: Title: Mean Field Method Applied to Lattice Gauge Theories with Compact Gauge Group Destination: section.6 Item: Title: Conclusions Destination: section.7 Item: Title: References Destination: section.8 Info: Title: Mean Field Approaches to Lattice Gauge Theories: A Review Author: Pierpaolo Fontana and Andrea Trombettoni Subject: Due to their broad applicability, gauge theories (GTs) play a crucial role in various areas of physics, from high-energy physics to condensed matter. Their formulations on lattices, lattice gauge theories (LGTs), can be studied, among many other methods, with tools coming from statistical mechanics lattice models, such as mean field methods, which are often used to provide approximate results. Applying these methods to LGTs requires particular attention due to the intrinsic local nature of gauge symmetry, how it is reflected in the variables used to formulate the theory, and the breaking of gauge invariance when approximations are introduced. This issue has been addressed over the decades in the literature, yielding different conclusions depending on the formulation of the theory under consideration. In this article, we focus on the mean field theoretical approach to the analysis of GTs and LGTs, connecting both older and more recent results that, to the best of our knowledge, have not been compared in a pedagogical manner. After a brief overview of mean field theory in statistical mechanics and many-body systems, we examine its application to pure LGTs with a generic compact gauge group. Finally, we review the existing literature on the subject, discussing the results obtained so far and their dependence on the formulation of the theory. 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