Lagrangian (field theory, $S[\varphi ]=\int d^{d}x\mathcal{L}(\varphi ,\partial \varphi ,x)$)

Lagrangian field theory specifies an action by integrating a Lagrangian density $\mathcal{L}(\varphi ,\partial \varphi ,x)$ over spacetime. Requiring its first variation to vanish for compactly supported field variations yields the field Euler-Lagrange equations. The variational Lagrangian is the integrand, whereas this field-theoretic formalism specifies the continuum fields and their dynamics.

A Lagrangian density assigns a local action contribution to field values and their derivatives. Integrating over spacetime gives $S=\int d^{d}x\mathcal{L}$. For a real scalar in flat spacetime with signature $(+---)$, a simple density is $\mathcal{L}=\frac{1}{2}{\partial }_{\mu }\varphi {\partial }^{\mu }\varphi -V(\varphi )$.

Varying the field gives $\square \varphi +V^{\prime }(\varphi )=0$, illustrating how the density encodes propagation and forces. Continuous symmetries yield Noether currents under the appropriate variational assumptions. The Lagrangian density is not generally the energy density: temporal and spatial derivative terms combine differently in the Hamiltonian. Adding a total derivative leaves the bulk equations unchanged under compatible boundary conditions, though boundary contributions and canonical expressions can change.

A kinetic term contains derivatives of a field and determines its free propagation. Its quadratic differential operator becomes the inverse quantum field theory propagator.

A gauge-covariant kinetic term replaces ordinary derivatives by covariant derivatives. Expanding it generates interactions between the matter field and one or more gauge fields, including contact vertices with two gauge bosons.

A mass term is a derivative-free quadratic field contribution whose coefficient fixes the linearized mass about a chosen background. For a real scalar with canonical kinetic normalization, $\mathcal{L}\supset -\frac{1}{2}{m}^2{\varphi }^2$ gives $(\square +m^2)\varphi =0$ and the relativistic dispersion $E^2={\mathbf{p}}^2+m^2$ in natural units.

Around an interacting vacuum, the mass squared comes from the potential curvature together with the kinetic normalization, not from every quadratic coefficient viewed in isolation. Several mixed fields require diagonalizing the corresponding quadratic problem. A negative curvature signals instability of that background toward a different configuration; it does not describe a stable particle simply traveling faster than light. The background and sign convention are therefore part of the statement.

The parameter multiplying a scalar quadratic mass term. A negative value describes an instability of the expansion point, rather than a negative value of the physical mass of a stable excitation. Nonnegative quartic terms do not stabilize a negative quadratic term along an exactly flat quartic ray.

A field interaction term is part of the action coupling fluctuations beyond a chosen free reference theory. Cubic and quartic scalar terms provide standard examples; in a perturbative expansion their coefficients and derivatives determine the corresponding interaction vertex.

The split into free and interacting parts is a calculational choice. Quadratic mixing can be diagonalized into a new free basis or treated as an insertion relative to another basis, so “nonlinear” alone is not a universal definition of every interaction term. Gauge-covariant kinetic terms generate derivative interactions and two-gauge-field contact terms when expanded. Specifying the complete reference action, field normalization and interaction coefficients makes the resulting perturbation theory reproducible.

A coupling constant is a coefficient of a field interaction term. It fixes the interaction's strength relative to the chosen field normalization. For example, a real scalar field can haveHere the factor $4!$ is a convention in the definition of $g$; changing that convention changes the parameter's numerical value. Its engineering dimension depends on the spacetime dimension and the normalization of the kinetic term.

In a renormalized theory, a coupling constant is specified at a renormalization scale $\mu$. Its change with that scale is described by the renormalization-group beta function, $\beta (g)=\mu dg/d\mu$. Thus the word “constant” need not mean independence of the scale used to describe the interaction.

The Euler-Lagrange field equation implements the principle of stationary action for a Lagrangian density depending on scalar fields and their first derivatives. For a single scalar field $\Phi$,

Vary $S=\int d^{4}x\mathcal{L}$. Integration by parts transfers the derivative from $\delta \Phi$ to $\partial \mathcal{L}/\partial ({\partial }_{\mu }\Phi )$. The boundary term vanishes for variations supported away from the boundary; arbitrary interior variations then force the displayed coefficient to vanish. For $\mathcal{L}=\frac{1}{2}{\partial }_{\mu }\Phi {\partial }^{\mu }\Phi -\frac{1}{2}{m}^2{\Phi }^2$, the result is the Klein-Gordon equation $({\partial }_{\mu }{\partial }^{\mu }+m^2)\Phi =0$, using natural units and Minkowski metric signature $(+---)$. ^[ref]

An off shell field configuration is not required to satisfy the Euler-Lagrange field equation. An external four-momentum is off shell if it is not constrained to its physical mass shell. On shell identities can therefore fail for general untransformed correlation functions, even when the associated S-matrix is unchanged by a field redefinition.

Adding $({\varphi }_1{\varphi }_{2,t}-{\varphi }_2{\varphi }_{1,t})/(2\tau )$ to the cross-coupled scalar Lagrangian density produces independent equations ${\varphi }_{1,tt}-c^2{\varphi }_{1,xx}+{\varphi }_{1,t}/\tau =0$ and ${\varphi }_{2,tt}-c^2{\varphi }_{2,xx}-{\varphi }_{2,t}/\tau =0$. For the convention $e^{i(kx-\omega t)}$, their frequencies are respectively $-i/(2\tau )\pm \sqrt{c^2{k}^2-1/(4{\tau }^2)}$ and $+i/(2\tau )\pm \sqrt{c^2{k}^2-1/(4{\tau }^2)}$. Above the threshold they oscillate with decaying or growing envelopes; below it they have nonoscillatory real temporal exponents. At equality the repeated temporal root supplies $(A+Bt)e^{\mp t/(2\tau )}$. At $k=0$ each equation also has a constant solution. The full system is generically unstable because of the amplified second field, even though it arises from an autonomous variational system with conserved, indefinite total energy.

For an action $S=\int \mathcal{L}(\varphi ,\partial \varphi ,{\partial }^2\varphi )d^{d+1}x$, vary the field with compact support, or fix both $\delta \varphi$ and its first derivatives on the boundary. One integration by parts on the first-derivative term and two on the second-derivative term give the displayed Euler-Lagrange field equation. With ordered repeated indices, define the second-derivative coefficient symmetrically so that $\delta \mathcal{L}={\mathcal{L}}_{\varphi }\delta \varphi +{\mathcal{L}}_{{\varphi }_{,\mu }}{\partial }_{\mu }\delta \varphi +{\mathcal{L}}_{{\varphi }_{,\mu \nu }}{\partial }_{\mu }{\partial }_{\nu }\delta \varphi$. This convention avoids an unintended factor of two in mixed derivative terms. Higher spatial derivatives alone do not require higher time derivatives.

A classical field configuration is on shell when it satisfies the specified Euler-Lagrange field equations. A relation holding only after those equations are used is an on-shell identity; an off-shell identity holds for arbitrary admissible configurations without imposing the equations.

For a free plane wave of a massive scalar, the field equation gives $p^2=m^2$. This connects the classical equation meaning to the on-shell momentum condition for a free external particle. Internal propagator momenta in a Feynman calculation need not obey that mass-shell relation, even though conservation at vertices remains enforced. Conserved Noether currents are generally conserved on shell, whereas some differential identities follow independently from the equations of motion.