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\begin{document}

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\section*{theory (physics)}

\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{general_idea}{General idea}\dotfill \pageref*{general_idea} \linebreak
\noindent\hyperlink{formalization}{Formalization}\dotfill \pageref*{formalization} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{general_principles}{General principles}\dotfill \pageref*{general_principles} \linebreak
\noindent\hyperlink{types_of_theories}{Types of theories}\dotfill \pageref*{types_of_theories} \linebreak
\noindent\hyperlink{the_fundamental_phenomenological_theories}{The fundamental phenomenological theories}\dotfill \pageref*{the_fundamental_phenomenological_theories} \linebreak
\noindent\hyperlink{TheoriesAndTheirModels}{Theories and their models}\dotfill \pageref*{TheoriesAndTheirModels} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

\hypertarget{general_idea}{}\subsubsection*{{General idea}}\label{general_idea}

In [[physics]] the term \emph{theory} or \emph{physical theory} traditionally refers, somewhat vaguely, to a given set of notions and rules, usually formulated in the language of [[mathematics]], that describe how some [[physical system]] or class of physical systems behaves. Typically these systems are highly idealized, in that the theories describe only certain aspects.

Often a given such theory depends on many free parameters. When a choice of such parameters is made or the range of the parameters is being restricted one tends to call the result a \emph{[[model (in theoretical physics)]]}. For more on this see \emph{\hyperlink{TheoriesAndTheirModels}{Theories and their Models}} below.

The most accurate general theory of fundamental physics known is \emph{[[Einstein gravity]]} and \emph{[[quantum field theory]]}. The best available choices of parameters in this general theory that make it fit the specifics of the observed world ([[phenomenology]]) are [[model (in theoretical physics)|models]] which accordingly are called the \emph{standard models}: there is the \emph{[[standard model of particle physics]]} and the \emph{[[standard model of cosmology]]}.

\hypertarget{formalization}{}\subsubsection*{{Formalization}}\label{formalization}

Beware that, therefore, the use of the terms \emph{theory} and \emph{model} in [[physics]] is \emph{different} from the same terms as used in [[logic]] (see at [[theory (logic)]] and [[model (logic)]]).

But most theories of fundamental physics (and many theories of effective physics such as [[solid state physics]]) fit into a pattern that can be [[axiom|axiomatized]] at least to some extent:

these physical theories are specified by a ([[local Lagrangian|local]]/[[extended Lagrangian|extended]]) [[Lagrangian]] on a space of [[field (physics)|fields]] over a given [[spacetime]]/[[worldvolume]] [[manifold]] $X$, hence by an [[action functional]]

\begin{displaymath}
S \;\colon\; [X, \mathbf{Fields}]_{\mathbf{H}} \to U(1)
  \,.
\end{displaymath}
In particular the corresponding [[classical field theory]] has as its ``space of models'' the [[critical locus]]

\begin{displaymath}
\underset{\phi \in [X, \mathbf{Fields}]_{\mathbf{H}}}{\sum}
  ( \mathbf{d} S_\phi \simeq 0 )
\end{displaymath}
of such an [[action functional]], the space of solutions of the [[Euler-Lagrange equations]]. A point in this space is a single ``physically realizable'' configuration of [[field (physics)|fields]] in this theory, disregarding [[quantum field theory]]-corrections, and a small-parameter subspace is often referred to as ``a model'' of the theory.

In this perspective of classical field theory, two different action functionals on different spaces of fields but with [[equivalence|equivalent]] [[critical loci]] are regarded as ``equivalent physical theories''. One often sees the term ``classically equivalent'' for this notion used in the literature.

But the full [[quantum field theory]] determined by a [[Lagrangian]]/[[action functional]] depends on more than just the [[critical locus]], which is just something like the lowest order approximation to the quantum theory (in a sense that can be made precise, for instance in [[deformation quantization]] in terms of [[power series]] developments in [[Planck's constant]].)

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{general_principles}{}\subsubsection*{{General principles}}\label{general_principles}

One broad way of classifying physical theories is by the extent to which they take [[quantum physics]] into account. We have

\begin{itemize}%
\item [[classical field theory]]


\item [[prequantum field theory]]


\item [[quantum field theory]]



\end{itemize}
Here [[quantum field theory]] is the most refined framework, which underlies the [[standard model of particle physics]].

The notion of quantum field theory, fundamental as it is, is quite flexible and in particular it naturally captures the concept that a given quantum field theory only describes phenomena that occur below a certain [[energy]] range and treats all phenomena at higher energy as the average over an unspecified more refined theory. This is the notion of

\begin{itemize}%
\item [[effective quantum field theory]].

\end{itemize}
Crucially, [[Einstein gravity]] is not known to have a formulation as a \emph{fundamental} quantum field theory with finitely many unspecified parameters ([[renormalization|renormalizable]]). But it may well be a [[effective quantum field theory]], the approximation to a more refined physical theory valid at higher energies. (This is the issue of [[quantum gravity]].) A proposal for a physical theory that achieves this is called \emph{[[string theory]]}.

\hypertarget{types_of_theories}{}\subsubsection*{{Types of theories}}\label{types_of_theories}

\begin{itemize}%
\item [[free field theory]]


\item [[gauge theory]]



\end{itemize}
\hypertarget{the_fundamental_phenomenological_theories}{}\subsubsection*{{The fundamental phenomenological theories}}\label{the_fundamental_phenomenological_theories}

\begin{itemize}%
\item [[gravity]], [[Yang-Mills theory]]


\item [[Einstein-Maxwell theory]]


\item [[Einstein-Yang-Mills theory]]


\item [[Einstein-Maxwell-Yang-Mills-Dirac-Higgs theory]]



\end{itemize}
[[!include standard model of fundamental physics - table]]

\hypertarget{TheoriesAndTheirModels}{}\subsubsection*{{Theories and their models}}\label{TheoriesAndTheirModels}

\begin{example}
\label{}\hypertarget{}{}
The [[classical field theory]] of [[gravity]] is a physical theory which asserts that [[spacetime]] is modeled by a [[pseudo-Riemannian manifold]] equipped with certain further [[force]] and [[matter]] [[field (physics)|fields]], such that this data satisfies [[Einstein equations]]. But if one furthermore specifies a particular such pseudo-Riemannian manifold etc. one may call this a \emph{model} of [[gravity]]/[[cosmology]]. The \emph{[[FRW model]]} is an example: here one specifies that the given pseudo-Riemannian metric and the [[matter]] [[field (physics)|field]] content is \emph{homogenous} and \emph{isotropic}. This is highly restrictive but still does not single out a unique solution. The remaining parameter is $k \in \{-1,0,1\}$, determining whether in this solution space has negative, positive or vanishing constant curvature.

\end{example}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[model (physics)]], [[experiment]], [[coordination]]


\item [[theory of everything]]


\item [[computable physics]]


\item [[mathematical physics]]


\item \href{http://ncatlab.org/nlab/show/string+theory+FAQ#AsideHowToPhysicalTheorieyGenerallyMakePredictionsAnyway}{string theory FAQ -- How do physical theories generally make predictions, anyway?}



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[James Wells]], \emph{Lexicon of Theory Qualities} (\href{http://www-personal.umich.edu/~jwells/prms/prm8.pdf}{pdf})

\end{itemize}
[[!redirects theories (physics)]]

[[!redirects physical theory]] [[!redirects physical theories]]

[[!redirects theory in physics]] [[!redirects theories in physics]]

[[!redirects theory of physics]] [[!redirects theories of physics]]



\end{document}