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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{lattice gauge theory} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{physics}{}\paragraph*{{Physics}}\label{physics} [[!include physicscontents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{SignProblem}{Sign problem}\dotfill \pageref*{SignProblem} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak \noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak \noindent\hyperlink{ReferencesMontoCarloSimulations}{Computer simulations}\dotfill \pageref*{ReferencesMontoCarloSimulations} \linebreak \noindent\hyperlink{renormalization}{Renormalization}\dotfill \pageref*{renormalization} \linebreak \noindent\hyperlink{topological_effects_and_instantons}{Topological effects and instantons}\dotfill \pageref*{topological_effects_and_instantons} \linebreak \noindent\hyperlink{ForSuperYangMills}{For super Yang-Mills theories}\dotfill \pageref*{ForSuperYangMills} \linebreak \noindent\hyperlink{general_2}{General}\dotfill \pageref*{general_2} \linebreak \noindent\hyperlink{ReferencesBFSS}{Compactification to $D = 1$}\dotfill \pageref*{ReferencesBFSS} \linebreak \noindent\hyperlink{ReferencesIKKT}{Compactification to $D= 0$}\dotfill \pageref*{ReferencesIKKT} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} \emph{Lattice gauge theory} (introduced in \hyperlink{Wilson74}{Wilson 74}) is [[gauge theory]] ([[Yang-Mills theory]], such as [[quantum chromodynamics]]) where [[continuum]] [[spacetime]] is replaced by a [[discrete group|discrete]] [[lattice (in a vector space, etc.)|lattice]], hence a [[lattice model]] for [[gauge field theory]]. Usually this is considered after [[Wick rotation]] from [[Minkowski spacetime]] $\mathbb{R}^{3,1}$ to [[Euclidean field theory]] on a [[lattice (in a vector space, etc.)|lattice]] inside $\mathbb{R}^3 \times S^1$, and typically one further identifies the spatial directions periodically to arrive at [[Euclidean]] [[gauge field theory]] on a [[lattice (in a vector space, etc.)|lattice]] inside the [[4-torus]] $T^4$. This discretization and further [[KK-compactification|compactification]] has the effect that the would-be [[path integral]] of the theory becomes an ordinary [[finite number|finite]]- (albeit high-)[[dimension|dimensional]] [[integral]], hence well defined and in principle amenable to explicit computation. This allows to consider ([[Wick rotation|Wick-rotated]]) [[path integral quantization]] at fixed lattice spacing, this being, in principle, a [[non-perturbative field theory|non-perturbative]] [[quantization]], in contrast to [[perturbative quantum field theory]] in terms of a [[Feynman perturbation series]]. On the other hand, much of the subtlety of the latter now appears in issues of taking the continuum limit where the the lattice spacing is sent to zero. In particular, different choices of discretizing the [[path integral]] over the lattice correspond to the [[renormalization]]-freedom seen in [[perturbative quantum field theory]]. Hence lattice gauge theory lends itself to brute-force simulation of [[quantum field theory]] on electronic computers, and the term is often understood by default in this sense. See \hyperlink{FodorHoelbling12}{Fodor-Hoelbling 12} for a good account. Since the explicit [[non-perturbative quantum field theory|non-perturbative]] formulation of [[Yang-Mills theories]] such as [[QCD]] is presently wide open (see the references at [[mass gap]] and at \emph{[[quantization of Yang-Mills theory]]}) these numerical simulation provide, besides actual [[experiment]], key insight into the non-perturbative nature of the theory, such as its [[instanton sea]] (\hyperlink{Gruber13}{Gruber 13}) and notably the phenomenonon of [[confinement]]/[[mass gap]] and explicit computation of [[hadron]] [[masses]] (\hyperlink{Durr09}{Durr et al. 09}, see \hyperlink{FodorHoelbling12}{Fodor-Hoelbling 12, section V}) Despite the word ``theory'', lattice gauge theory is more like ``computer-simpulated [[experiment]]''. While it allows to see phenomena of QCD, it usually cannot provide a conceptual explanation, and of course not a mathematical derivation of problems such as [[confinement]]/[[mass gap]]. Lattice gauge theory is to the [[confinement]]/[[mass gap]]-problems as explicit computation of zeros of the [[Riemann zeta-function]] is to the [[Riemann hypothesis]] (see \hyperlink{Riemann+hypothesis#ReferencesComputerChecks}{there})). \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{SignProblem}{}\subsubsection*{{Sign problem}}\label{SignProblem} (\ldots{}) \begin{itemize}% \item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Numerical_sign_problem}{Numerical sign problem}} \end{itemize} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[lattice model]] \item [[non-perturbative effect]] \item [[lattice renormalization]] \item Discussion of [[QCD instantons]] in LGT includes (\hyperlink{Moore03}{Moore 03, section 7}, \hyperlink{Gruber13}{Gruber 13}) \item [[AdS/QCD correspondence]] \item [[Euclidean field theory]] \item [[string bit model]] \end{itemize} \hypertarget{References}{}\subsection*{{References}}\label{References} The concept was introduced in \begin{itemize}% \item [[Kenneth Wilson]], \emph{Confinement of quarks, Phys. Rev. D10, 2445, 1974 (\href{https://doi.org/10.1103/PhysRevD.10.2445}{doi:10.1103/PhysRevD.10.2445})} \end{itemize} \hypertarget{general}{}\subsubsection*{{General}}\label{general} \begin{itemize}% \item S. Gupta, \emph{Introduction to lattice field theory}, March 2011, (\href{http://theory.tifr.res.in/~sgupta/talks/11aslft.pdf}{pdf}) \item G. M\"u{}nster, M. Walzl, \emph{Lattice Gauge Theory - A short Primer} (\href{http://arxiv.org/abs/hep-lat/0012005}{arXiv:hep-lat/0012005}) \item [[Kenneth Wilson]], \emph{The Origins of Lattice Gauge Theory}, (\href{http://arxiv.org/abs/hep-lat/0412043}{arXiv:hep-lat/0412043}) \item Guy Moore, \emph{Informal lectures on lattice gauge theory}, 2003 (\href{https://theorie.ikp.physik.tu-darmstadt.de/qcd/moore/latt_lectures.pdf}{pdf}) \item [[Kasper Peeters]], [[Marija Zamaklar]], section 5 of \emph{Euclidean Field Theory}, Lecture notes 2009-2011 (\href{http://maths.dur.ac.uk/users/kasper.peeters/eft.html}{web}, \href{http://maths.dur.ac.uk/users/kasper.peeters/pdf/eft.pdf}{pdf}) \end{itemize} Visualization: \begin{itemize}% \item James Biddle et al. \emph{Publicising Lattice Field Theory through Visualisation} (\href{https://arxiv.org/abs/1903.08308}{arXiv:1903.08308}) \end{itemize} Relation to [[string theory]]/[[M-theory]] (such as via [[BFSS matrix model]]) in view [[AdS-CFT duality]]: \begin{itemize}% \item Masanori Hanada, \emph{What lattice theorists can do for superstring/M-theory}, International Journal of Modern Physics AVol. 31, No. 22, 1643006 (2016) (\href{https://arxiv.org/abs/1604.05421}{arXiv:1604.05421}) \end{itemize} See also \begin{itemize}% \item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Lattice_gauge_theory}{Lattice gauge theory}} \item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Lattice_QCD}{Lattice QCD}} \end{itemize} \hypertarget{ReferencesMontoCarloSimulations}{}\subsubsection*{{Computer simulations}}\label{ReferencesMontoCarloSimulations} A good general account of computer simulation of lattice [[QCD]] is in \begin{itemize}% \item Zoltan Fodor, Christian Hoelbling, sections II-IV of \emph{Light Hadron Masses from Lattice QCD}, Rev. Mod. Phys. 84, 449 (2012) (\href{https://arxiv.org/abs/1203.4789}{arXiv:1203.4789}) \end{itemize} See also \begin{itemize}% \item [[Michael Creutz]], \emph{Monte Carlo study of quantized SU(2) gauge theory} Phys. Rev. D21 (1980) 2308-2315 (\href{http://prd.aps.org/abstract/PRD/v21/i8/p2308_1}{journal}, \href{http://thy.phy.bnl.gov/~creutz/mypubs/pub037.pdf}{pdf}) \item [[Michael Creutz]], \emph{Monte Carlo study of renormalization in lattice gauge theory} Phys.Rev. D23 (1981) 1815 (\href{http://thy.phy.bnl.gov/~creutz/mypubs/pub045.pdf}{pdf}) \item [[Michael Creutz]], Laurence Jacobs, Claudio Rebbi, \emph{Monte Carlo computations in lattice gauge theories}, Volume 95, Issue 4, April 1983, Pages 201--282 (\href{http://thy.phy.bnl.gov/~creutz/mypubs/pub068.pdf}{pdf}) \end{itemize} Specifically computation of [[hadron]]-[[masses]] (see [[mass gap problem]]) in lattice QCD is reported here: \begin{itemize}% \item S. Durr, Z. Fodor, J. Frison, C. Hoelbling, R. Hoffmann, S.D. Katz, S. Krieg, T. Kurth, L. Lellouch, T. Lippert, K.K. Szabo, G. Vulvert, \emph{Ab-initio Determination of Light Hadron Masses}, Science 322:1224-1227,2008 (\href{https://arxiv.org/abs/0906.3599}{arXiv:0906.3599}) \end{itemize} reviewed in \begin{itemize}% \item \hyperlink{FodorHoelbling12}{Fodor-Hoelbling 12, section V of} \end{itemize} \hypertarget{renormalization}{}\subsubsection*{{Renormalization}}\label{renormalization} A proposal for a rigorous formulation of [[renormalization]] in lattice gauge theory is due to \begin{itemize}% \item [[Tadeusz Balaban]], \emph{Renormalization group approach to lattice gauge field theories: I. Generation of effective actions in a small field approximation and a coupling constant renormalization in four dimensions}, Communications in Mathematical Physics, Volume 109, Issue 2, pp.249-301 (\href{https://link.springer.com/article/10.1007%2FBF01215223}{web}) \item \ldots{} \end{itemize} reviewed in \begin{itemize}% \item [[Jonathan Dimock]], \emph{The renormalization group according to Balaban, I. Small fields}, Rev. Math. Phys., 25, 1330010 (2013) (\href{https://doi.org/10.1142/S0129055X13300100}{doi:10.1142/S0129055X13300100}) \item \ldots{} \end{itemize} \hypertarget{topological_effects_and_instantons}{}\subsubsection*{{Topological effects and instantons}}\label{topological_effects_and_instantons} Discussion of [[instantons]] in lattice QCd \begin{itemize}% \item Florian Gruber, \emph{Topology in dynamical Lattice QCD simulations}, 2013 (\href{http://epub.uni-regensburg.de/27631/}{web}, \href{http://epub.uni-regensburg.de/27631/1/dissertation.pdf}{pdf}) \end{itemize} \hypertarget{ForSuperYangMills}{}\subsubsection*{{For super Yang-Mills theories}}\label{ForSuperYangMills} Lattice simulation of [[torus]]-[[KK-compactifications]] of [[10d super Yang-Mills theory]] and numerical test of [[AdS/CFT]]: \hypertarget{general_2}{}\paragraph*{{General}}\label{general_2} \begin{itemize}% \item Anosh Joseph, \emph{Review of Lattice Supersymmetry and Gauge-Gravity Duality} (\href{https://arxiv.org/abs/1509.01440}{arXiv:1509.01440}) \item Masanori Hanada, \emph{What lattice theorists can do for superstring/M-theory}, International Journal of Modern Physics AVol. 31, No. 22, 1643006 (2016) (\href{https://arxiv.org/abs/1604.05421}{arXiv:1604.05421}) \end{itemize} \hypertarget{ReferencesBFSS}{}\paragraph*{{Compactification to $D = 1$}}\label{ReferencesBFSS} The [[BFSS matrix model]]: \begin{itemize}% \item Veselin G. Filev, Denjoe O'Connor, \emph{The BFSS model on the lattice}, JHEP 1605 (2016) 167 (\href{https://arxiv.org/abs/1506.01366}{arXiv:1506.01366}) \item Masanori Hanada, Paul Romatschke, \emph{Lattice Simulations of 10d Yang-Mills toroidally compactified to 1d, 2d and 4d} (\href{https://arxiv.org/abs/1612.06395}{arXiv:1612.06395}) \end{itemize} \hypertarget{ReferencesIKKT}{}\paragraph*{{Compactification to $D= 0$}}\label{ReferencesIKKT} The [[IKKT matrix model]] and claims that it predicts spontaneous [[KK-compactification]] of the $D = 10$ [[M-theory|non-perturbative]] [[type IIB string theory]]/[[F-theory]] to $D = 3+1$ macrocopic [[spacetime]] [[dimensions]]: \begin{itemize}% \item S.-W. Kim, J. Nishimura, and A. Tsuchiya, \emph{Expanding (3+1)-dimensional universe from a Lorentzian matrix model for superstring theory in (9+1)-dimensions}, Phys. Rev. Lett. 108, 011601 (2012), (\href{https://arxiv.org/abs/1108.1540}{arXiv:1108.1540}). \item S.-W. Kim, J. Nishimura, and A. Tsuchiya, \emph{Late time behaviors of the expanding universe in the IIB matrix model}, JHEP 10, 147 (2012), (\href{https://arxiv.org/abs/1208.0711}{arXiv:1208.0711}). \item Yuta Ito, Jun Nishimura, Asato Tsuchiya, \emph{Large-scale computation of the exponentially expanding universe in a simplified Lorentzian type IIB matrix model} (\href{https://arxiv.org/abs/1512.01923}{arXiv:1512.01923}) \item Toshihiro Aoki, Mitsuaki Hirasawa, Yuta Ito, Jun Nishimura, Asato Tsuchiya, \emph{On the structure of the emergent 3d expanding space in the Lorentzian type IIB matrix model} (\href{https://arxiv.org/abs/1904.05914}{arXiv:1904.05914}) \end{itemize} [[!redirects lattice gauge theories]] [[!redirects lattice gauge field theory]] [[!redirects lattice gauge field theorues]] [[!redirects lattice field theory]] [[!redirects lattice field theories]] [[!redirects lattice QCD]] \end{document}