Editing Classical field theory (section)

== Relativistic field theory ==
{{Main|Covariant classical field theory}}

Modern formulations of classical field theories generally require [[Lorentz covariance]] as this is now recognised as a fundamental aspect of nature. A field theory tends to be expressed mathematically by using [[Lagrangian (field theory)|Lagrangian]]s. This is a function that, when subjected to an [[action principle]], gives rise to the [[field equations]] and a [[Conservation law (physics)|conservation law]] for the theory. The [[action (physics)|action]] is a Lorentz scalar, from which the field equations and symmetries can be readily derived.

Throughout we use units such that the speed of light in vacuum is 1, i.e. ''c'' = 1.{{NoteTag|This is equivalent to choosing units of distance and time as light-seconds and seconds or light-years and years. Choosing ''c'' {{=}} 1 allows us to simplify the equations. For instance, ''E'' {{=}} ''mc''<sup>2</sup> reduces to ''E'' {{=}} ''m'' (since ''c''<sup>2</sup> {{=}} 1, without keeping track of units). This reduces complexity of the expressions while keeping focus on the underlying principles. This "trick" must be taken into account when performing actual numerical calculations.}}

=== Lagrangian dynamics ===
{{Main|Lagrangian (field theory)}}

Given a field tensor <math>\phi</math>, a scalar called the [[Lagrangian density]] <math display="block">\mathcal{L}(\phi,\partial\phi,\partial\partial\phi, \ldots ,x)</math> can be constructed from <math>\phi</math> and its derivatives.
From this density, the action functional can be constructed by integrating over spacetime,
<math display="block">\mathcal{S} = \int{\mathcal{L}\sqrt{-g}\, \mathrm{d}^4x}.</math>

Where <math>\sqrt{-g} \, \mathrm{d}^4x</math> is the volume form in curved spacetime. <math>(g\equiv \det(g_{\mu\nu}))</math>

Therefore, the Lagrangian itself is equal to the integral of the Lagrangian density over all space.

Then by enforcing the [[Action (physics)|action principle]], the Euler–Lagrange equations are obtained

<math display="block">\frac{\delta \mathcal{S}}{\delta\phi} = \frac{\partial\mathcal{L}}{\partial\phi} -\partial_\mu \left(\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\right)+ \cdots +(-1)^m\partial_{\mu_1} \partial_{\mu_2} \cdots \partial_{\mu_{m-1}} \partial_{\mu_m} \left(\frac{\partial\mathcal{L}}{\partial(\partial_{\mu_1} \partial_{\mu_2}\cdots\partial_{\mu_{m-1}}\partial_{\mu_m} \phi)}\right) = 0.</math>