MathematicalStructureOfPhysicalTheories 2009-03-25 13:42:52 2026-08-23 02:13:26 Topic classical mechanics Newtonian mechanics relativity theories quantum theories wave theories QED QCD QFT AQFT TQFT HQFT electroweak theories gravitational theories quantum gravity string and superstring theories high energy physics theories supersymmetry SUSY Astrophysics theories chaos theories complex systems biophysics mathematical biophysics theories categorical physics gauge theories renormalizable theories non-renormalizable physical theories conformal theories statistical physics calculus and analytical geometry differential geometry noncommutative geometry quantum geometry Riemanian and pseudo-Riemannian manifolds Hilbert spaces C*-algebra Clifford algebra quantum operator theory and functional analysis probability theory statistics harmonic and anharmonic analysis Lebesgue measure theory Haar systems non-Abelian algebraic topology group theory topology and groupoids category theory abstract algebra matrix algebra superstructures supergroups superalgebras Lie groups and Lie algebras graded Lie algebras algebroids rings and modules algebraic geometry quantum logics axiomatics many-valued logics Boolean logic automata and dynamic system theories complex systems biphysics % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amsthm} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \newcommand{\sR}[0]{\mathbb{R}} \newcommand{\sC}[0]{\mathbb{C}} \newcommand{\sN}[0]{\mathbb{N}} \newcommand{\sZ}[0]{\mathbb{Z}} \usepackage{bbm} \newcommand{\Z}{\mathbbmss{Z}} \newcommand{\C}{\mathbbmss{C}} \newcommand{\R}{\mathbbmss{R}} \newcommand{\Q}{\mathbbmss{Q}} \newcommand*{\norm}[1]{\lVert #1 \rVert} \newcommand*{\abs}[1]{| #1 |} \newtheorem{thm}{Theorem} \newtheorem{defn}{Definition} \newtheorem{prop}{Proposition} \newtheorem{lemma}{Lemma} \newtheorem{cor}{Corollary} This is a contributed topic on the mathematical structures associated with physical theories. Hopefully, this topic will be of interest to both physicists and mathematicians (as well as logicians and philosopy of science researchers). \section{Mathematical and Logical Foundations of Physics} \subsection{Physical Theories} \begin{itemize} \item Classical, or Newtonian, Mechanics: \PMlinkname{Kinematics}{NewtonsLawsOfMotion} and \PMlinkname{Dynamics}{NewtonsLawsOfMotion}, Netwon's Laws of Motion \item Classical Molecular Dynamics \item Classical Thermodynamics and Statistics \item Classical Electromagnetism: Maxwell's theory and Maxwell's Equations \item Optics and Spectroscopic Theories \item General and Special Relativity \item Quantum Mechanics \item Quantum Field Theories \item Mathematical Foundations of Quantum Mechanics \item Axiomatic Quantum Field Theories \item Local Quantum Physics theories \item Statistical Quantum Mechanics \item Standard Model (SUSY) \item Electroweak Theory \item Quantum Chromodynamics \item Quantum Gravity theories \item Strings and Branes theories \item Astrophysics theories \item Chaotic Dynamics and Complex Systems theories \item XYZ \end{itemize} \subsection{Mathematical structures and fields underlying physical theories} \begin{itemize} \item Calculus and Analytical Geometry \item Non-Newtonian Calculus \item Vector Calculus \item Functional Analysis \item Differential Geometry \item Algebra, Matrices and Tensors \item Category Theory \item Algebraic Topology and Non-Abelian Algebraic Theory \item Noncommutative Geometry \item Algebraic Geometry \item Harmonic and Anharmonic Analysis \item Numerical Analysis \item Statistics and Probability Theories \item Model Theory \item Superstructures: supergroups, superalgebras, algebraic supervarieties, super-categories \item XYZ \end{itemize} \subsection{Logical structure of physical theories} \begin{itemize} \item Axiomatics and Physical Postulates \item Quantum Logics \item Automata theory and Boolean logic designs \item Quantum Automata and Quantum Computers \item Many-valued Logic Algebra \end{itemize}