# Contents

## Idea

### General idea

In physics the term theory or 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 model (in theoretical physics). For more on this see Theories and their Models below.

The most accurate general theory of fundamental physics known is Einstein gravity and 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 models which accordingly are called the standard models: there is the standard model of particle physics and the standard model of cosmology.

### Formalization

Beware that, therefore, the use of the terms theory and model in physics is 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 axiomatized at least to some extent:

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

$S \;\colon\; [X, \mathbf{Fields}]_{\mathbf{H}} \to U(1) \,.$

In particular the corresponding classical field theory has as its “space of models” the critical locus

$\underset{\phi \in [X, \mathbf{Fields}]_{\mathbf{H}}}{\sum} ( \mathbf{d} S_\phi \simeq 0 )$

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 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 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.)

## Examples

### General principles

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

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

Crucially, Einstein gravity is not known to have a formulation as a fundamental quantum field theory with finitely many unspecified parameters (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 string theory.

### The fundamental phenomenological theories

standard model of particle physics and cosmology

theory:Einstein-Yang-Mills-Dirac-Higgs
gravityelectroweak and strong nuclear forcefermionic matterscalar field
field content:vielbein field $e$principal connection $\nabla$spinor $\psi$scalar field $H$
Lagrangian:scalar curvature densityfield strength squaredDirac operator component densityfield strength squared + potential density
$L =$$R(e) vol(e) +$$\langle F_\nabla \wedge \star_e F_\nabla\rangle +$$(\psi , D_{(e,\nabla)} \psi) vol(e) +$$\nabla \bar H \wedge \star_e \nabla H + \left(\lambda {\vert H\vert}^4 - \mu^2 {\vert H\vert}^2 \right) vol(e)$

### Theories and their models

###### Example

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 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 model of gravity/cosmology. The FRW model is an example: here one specifies that the given pseudo-Riemannian metric and the matter field content is homogenous and 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.