Lagrangian field theory is a formalism inclassical field theory. It is the field-theoretic analogue ofLagrangian mechanics. Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number ofdegrees of freedom. Lagrangian field theory applies to continua andfields, which have an infinite number of degrees of freedom.

One motivation for the development of the Lagrangian formalism on fields, and more generally, forclassical field theory, is to provide a clear mathematical foundation forquantum field theory, which is infamously beset by formal difficulties that make it unacceptable as a mathematical theory. The Lagrangians presented here are identical to their quantum equivalents, but, in treating the fields as classical fields, instead of being quantized, one can provide definitions and obtain solutions with properties compatible with the conventional formal approach to the mathematics ofpartial differential equations. This enables the formulation of solutions on spaces with well-characterized properties, such asSobolev spaces. It enables various theorems to be provided, ranging from proofs of existence to theuniform convergence of formal series to the general settings ofpotential theory. In addition, insight and clarity is obtained by generalizations toRiemannian manifolds andfiber bundles, allowing the geometric structure to be clearly discerned and disentangled from the corresponding equations of motion. A clearer view of the geometric structure has in turn allowed highly abstract theorems from geometry to be used to gain insight, ranging from theChern–Gauss–Bonnet theorem and theRiemann–Roch theorem to theAtiyah–Singer index theorem andChern–Simons theory.

In field theory, the independent variable is replaced by an event inspacetime(x,y,z,t), or more generally still by a points on aRiemannian manifold. The dependent variables are replaced by the value of a field at that point in spacetime${\displaystyle \phi (x,y,z,t)}$ so that theequations of motion are obtained by means of anaction principle, written as:$${\displaystyle \frac{\delta \mathcal{S}}{\delta {\phi }_i}=0,}$$where theaction,${\displaystyle \mathcal{S}}$, is afunctional of the dependent variables${\displaystyle {\phi }_i(s)}$, their derivatives ands itself

$${\displaystyle \mathcal{S}\left[{\phi }_i\right]=\int \mathcal{L}\left({\phi }_i(s),\left\{\frac{\mathrm{\partial }{\phi }_i(s)}{\mathrm{\partial }{s}^{\alpha }}\right\},\{s^{\alpha }\}\right){\mathrm{d}}^{n}s,}$$

where the brackets denote${\displaystyle \{\cdot \mathrm{\forall }\alpha \}}$;ands = {s^α} denotes theset ofnindependent variables of the system, including the time variable, and is indexed byα = 1, 2, 3, ...,n. The calligraphic typeface,${\displaystyle \mathcal{L}}$, is used to denote thedensity, and${\displaystyle {\mathrm{d}}^{n}s}$ is thevolume form of the field function, i.e., the measure of the domain of the field function.

In mathematical formulations, it is common to express the Lagrangian as a function on afiber bundle, wherein the Euler–Lagrange equations can be interpreted as specifying thegeodesics on the fiber bundle, leading to topics liketangent manifolds,symplectic manifolds andcontact geometry.^[1] The field theories of physics can be developed in terms of gauge invariant fiber bundles.^[2]

In Lagrangian field theory, the Lagrangian as a function ofgeneralized coordinates is replaced by a Lagrangian density, a function of the fields in the system and their derivatives, and possibly the space and time coordinates themselves. In field theory, the independent variablet is replaced by an event in spacetime(x,y,z,t) or still more generally by a points on a manifold.

Often, a "Lagrangian density" is simply referred to as a "Lagrangian".

For one scalar field${\displaystyle \phi }$, the Lagrangian density will take the form:^{[nb 1][3]}$${\displaystyle \mathcal{L}(\phi ,\mathbf{\nabla }\phi ,\mathrm{\partial }\phi /\mathrm{\partial }t,\mathbf{x},t)}$$

For many scalar fields$${\displaystyle \mathcal{L}({\phi }_1,\mathbf{\nabla }{\phi }_1,\mathrm{\partial }{\phi }_1/\mathrm{\partial }t,\dots ,{\phi }_n,\mathbf{\nabla }{\phi }_n,\mathrm{\partial }{\phi }_n/\mathrm{\partial }t,\dots ,\mathbf{x},t)}$$

In mathematical formulations, the scalar fields are understood to becoordinates on afiber bundle, and the derivatives of the field are understood to besections of thejet bundle.

The above can be generalized forvector fields,tensor fields, andspinor fields. In physics,fermions are described by spinor fields.Bosons are described by tensor fields, which include scalar and vector fields as special cases.

For example, if there are${\displaystyle m}$real-valuedscalar fields,${\displaystyle {\phi }_1,\dots ,{\phi }_m}$, then the field manifold is${\displaystyle {\mathbb{R}}^m}$. If the field is a realvector field, then the field manifold isisomorphic to${\displaystyle {\mathbb{R}}^n}$.

Thetime integral of the Lagrangian is called theaction denoted byS. In field theory, a distinction is occasionally made between theLagrangianL, of which the time integral is the action$${\displaystyle \mathcal{S}=\int L\mathrm{d}t,}$$and theLagrangian density${\displaystyle \mathcal{L}}$, which one integrates over allspacetime to get the action:$${\displaystyle \mathcal{S}[\phi ]=\int \mathcal{L}(\phi ,\mathbf{\nabla }\phi ,\mathrm{\partial }\phi /\mathrm{\partial }t,\mathbf{x},t){\mathrm{d}}^3\mathbf{x}\mathrm{d}t.}$$

The spatialvolume integral of the Lagrangian density is the Lagrangian; in 3D,$${\displaystyle L=\int \mathcal{L}{\mathrm{d}}^3\mathbf{x}.}$$

The action is often referred to as the "actionfunctional", in that it is a function of the fields (and their derivatives).

In the presence of gravity or when using general curvilinear coordinates, the Lagrangian density${\displaystyle \mathcal{L}}$ will include a factor of$\sqrt{g}$. This ensures that the action is invariant under general coordinate transformations. In mathematical literature, spacetime is taken to be aRiemannian manifold${\displaystyle M}$ and the integral then becomes thevolume form$${\displaystyle \mathcal{S}={\int }_M\sqrt{|g|}d{x}^1\wedge \cdots \wedge d{x}^m\mathcal{L}}$$

Here, the${\displaystyle \wedge }$ is thewedge product and$\sqrt{|g|}$ is the square root of the determinant${\displaystyle |g|}$ of themetric tensor${\displaystyle g}$ on${\displaystyle M}$. For flat spacetime (e.g.,Minkowski spacetime), the unit volume is one, i.e.$\sqrt{|g|}=1$ and so it is commonly omitted, when discussing field theory in flat spacetime. Likewise, the use of the wedge-product symbols offers no additional insight over the ordinary concept of a volume in multivariate calculus, and so these are likewise dropped. Some older textbooks, e.g., Landau and Lifschitz write$\sqrt{-g}$ for the volume form, since the minus sign is appropriate for metric tensors with signature (+−−−) or (−+++) (since the determinant is negative, in either case). When discussing field theory on general Riemannian manifolds, the volume form is usually written in the abbreviated notation${\displaystyle \ast (1)}$ where${\displaystyle \ast }$ is theHodge star. That is,$${\displaystyle \ast (1)=\sqrt{|g|}d{x}^1\wedge \cdots \wedge d{x}^m}$$and so$${\displaystyle \mathcal{S}={\int }_M\ast (1)\mathcal{L}}$$

Not infrequently, the notation above is considered to be entirely superfluous, and$${\displaystyle \mathcal{S}={\int }_M\mathcal{L}}$$is frequently seen. Do not be misled: the volume form is implicitly present in the integral above, even if it is not explicitly written.

TheEuler–Lagrange equations describe thegeodesic flow of the field${\displaystyle \phi }$ as a function of time. Taking thevariation with respect to${\displaystyle \phi }$, one obtains$${\displaystyle 0=\frac{\delta \mathcal{S}}{\delta \phi }={\int }_M\ast (1)\left(-{\mathrm{\partial }}_{\mu }\left(\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }({\mathrm{\partial }}_{\mu }\phi )}\right)+\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }\phi }\right).}$$

Solving, with respect to theboundary conditions, one obtains theEuler–Lagrange equations:$${\displaystyle \frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }\phi }={\mathrm{\partial }}_{\mu }\left(\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }({\mathrm{\partial }}_{\mu }\phi )}\right).}$$

Often the Lagrangian consists of a sum ofpolynomial terms, with thesymmetries of the theory and the fields involved dictating the types of terms that are allowed. For example, inrelativistic theories, each term must beLorentz invariant while in a theory with agauge field, they must be gauge invariant.

Terms that contain the product of two fields and noderivatives are known asmass terms, with these givingmass to the fields.^[4] For example, a single real scalar field${\displaystyle \varphi (x)}$ of mass${\displaystyle m}$ has a mass term given by

${\displaystyle {\mathcal{L}}_m=-\frac{1}{2}{m}^2{\varphi }^2(x).}$

The other terms that have two fields, those with at least one derivative, are known askinetic terms. They make fieldsdynamical, with most theories requiring a restriction of at most two derivatives in kinetic terms topreserve probabililties in aquantum theory. They are also usuallypositive-definite to ensure positive energies.^{[nb 2]} For example, the kinetic term for a relativistic real scalar field is given by

${\displaystyle {\mathcal{L}}_k=\frac{1}{2}{\mathrm{\partial }}_{\mu }\varphi {\mathrm{\partial }}^{\mu }\varphi .}$

Fields with no kinetic terms can also be found, playing the role ofauxiliary fields, background fields, orcurrents. Theories with only kinetic and mass terms, formfree field theories.

Any term with more than two fields per term is known as aninteraction term.^[5] The presence of these gives rise to interacting theories where particles canscatter off each other. The coefficients in front of these terms are known ascoupling constants and they dictate the strength of the interaction. For example, aquartic interaction in a real scalar field theory is given by

${\displaystyle {\mathcal{L}}_i=-\frac{g}{4!}{\varphi }^4,}$

where${\displaystyle g}$ is its coupling constant. This term gives rise to scattering processes whereby two scalar fields can scatter off each other. Interacting terms can have any number of derivatives, with each derivative providing amomentum dependence to the scattering term as can be seen by going intomomentum space.

Terms with only one field are known astadpole terms since they give rise totadpole Feynman diagrams.^[4]: 415 In theories withtranslational symmetries, such terms can usually be eliminated by redefining some of the fields though a shift.^{[nb 3]}

Constant terms, those with no fields, have no physical consequences in non-gravitational theories.^[6] In classical field theories, the equations of motion only depend on variations of the Lagrangian, so constant terms play no role. In quantum field theories they only provide an irrelevant overall multiplicative term to thepartition function, so again play no role. Physically this is because in these theories there is no absoluteenergy scale as thepotential energy can always be shifted by an arbitrary constant without altering the physics. However, ingravitational systems the constant terms are multiplied by the metric determinant, coupling them to the spacetime. They play the role of thecosmological constant, directly affecting the dynamics of the theory at both a classical and quantum level.

Polynomial terms are often expressed with certaincanonical normalizations, used to simplify theFeynman rules that are derived from them. Usually one divides by the product of thefactorial of themultipicity of the fields. For example, in a theory with two real scalar fields, a term of the form${\displaystyle g{\varphi }^n{\phi }^m}$ term would be divided by${\displaystyle n!m!}$. Particles andantiparticles are distinguished in this counting, so that a complex scalar field term of the form${\displaystyle g^{\prime }{\overline{\varphi }}^p{\varphi }^p}$ is divided by${\displaystyle p!p!}$ rather than${\displaystyle (2p)!}$.

A large variety of physical systems have been formulated in terms of Lagrangians over fields. Below is a sampling of some of the most common ones found in physics textbooks on field theory.

The Lagrangian density for Newtonian gravity is:

$${\displaystyle \mathcal{L}(\mathbf{x},t)=-\frac{1}{8\pi G}(\mathrm{\nabla }\mathrm{\Phi }(\mathbf{x},t){)}^2-\rho (\mathbf{x},t)\mathrm{\Phi }(\mathbf{x},t)}$$whereΦ is thegravitational potential,ρ is the mass density, andG in m³·kg⁻¹·s⁻² is thegravitational constant. The density${\displaystyle \mathcal{L}}$ has units of J·m⁻³. Here the interaction term involves a continuous mass densityρ in kg·m⁻³. This is necessary because using a point source for a field would result in mathematical difficulties.

This Lagrangian can be written in the form of${\displaystyle \mathcal{L}=T-V}$, with the${\displaystyle T=-(\mathrm{\nabla }\mathrm{\Phi }{)}^2/8\pi G}$ providing a kinetic term, and the interaction${\displaystyle V=\rho \mathrm{\Phi }}$ the potential term. See alsoNordström's theory of gravitation for how this could be modified to deal with changes over time. This form is reprised in the next example of a scalar field theory.

The variation of the integral with respect toΦ is:$${\displaystyle \delta \mathcal{L}(\mathbf{x},t)=-\rho (\mathbf{x},t)\delta \mathrm{\Phi }(\mathbf{x},t)-\frac{2}{8\pi G}(\mathrm{\nabla }\mathrm{\Phi }(\mathbf{x},t))\cdot (\mathrm{\nabla }\delta \mathrm{\Phi }(\mathbf{x},t)).}$$

After integrating by parts, discarding the total integral, and dividing out byδΦ the formula becomes:$${\displaystyle 0=-\rho (\mathbf{x},t)+\frac{1}{4\pi G}\mathrm{\nabla }\cdot \mathrm{\nabla }\mathrm{\Phi }(\mathbf{x},t)}$$which is equivalent to:$${\displaystyle 4\pi G\rho (\mathbf{x},t)={\mathrm{\nabla }}^2\mathrm{\Phi }(\mathbf{x},t)}$$which yieldsGauss's law for gravity.

The Lagrangian for a scalar field moving in a potential${\displaystyle V(\varphi )}$ can be written as$${\displaystyle \mathcal{L}=\frac{1}{2}{\mathrm{\partial }}^{\mu }\varphi {\mathrm{\partial }}_{\mu }\varphi -V(\varphi )=\frac{1}{2}{\mathrm{\partial }}^{\mu }\varphi {\mathrm{\partial }}_{\mu }\varphi -\frac{1}{2}{m}^2{\varphi }^2-\sum _{n=3}^{\mathrm{\infty }}\frac{1}{n!}{g}_n{\varphi }^n}$$It is not at all an accident that the scalar theory resembles the undergraduate textbook Lagrangian${\displaystyle L=T-V}$ for thekinetic term of a free point particle written as${\displaystyle T=m{v}^2/2}$. The scalar theory is the field-theory generalization of a particle moving in a potential. When the${\displaystyle V(\varphi )}$ is theMexican hat potential, the resulting fields are termed theHiggs fields.

Thesigma model describes the motion of a scalar point particle constrained to move on aRiemannian manifold, such as a circle or a sphere. It generalizes the case of scalar and vector fields, that is, fields constrained to move on a flat manifold. The Lagrangian is commonly written in one of three equivalent forms:$${\displaystyle \mathcal{L}=\frac{1}{2}\mathrm{d}\varphi \wedge \ast \mathrm{d}\varphi }$$where the${\displaystyle \mathrm{d}}$ is thedifferential. An equivalent expression is$${\displaystyle \mathcal{L}=\frac{1}{2}\sum _{i=1}^n\sum _{j=1}^n{g}_{ij}(\varphi ){\mathrm{\partial }}^{\mu }{\varphi }_i{\mathrm{\partial }}_{\mu }{\varphi }_j}$$with${\displaystyle g_{ij}}$ theRiemannian metric on the manifold of the field; i.e. the fields${\displaystyle {\varphi }_i}$ are justlocal coordinates on thecoordinate chart of the manifold. A third common form is$${\displaystyle \mathcal{L}=\frac{1}{2}\mathrm{t}\mathrm{r}\left(L_{\mu }{L}^{\mu }\right)}$$with$${\displaystyle L_{\mu }=U^{-1}{\mathrm{\partial }}_{\mu }U}$$and${\displaystyle U\in \mathrm{S}\mathrm{U}(N)}$, theLie groupSU(N). This group can be replaced by any Lie group, or, more generally, by asymmetric space. The trace is just theKilling form in hiding; the Killing form provides a quadratic form on the field manifold, the lagrangian is then just the pullback of this form. Alternately, the Lagrangian can also be seen as the pullback of theMaurer–Cartan form to the base spacetime.

In general, sigma models exhibittopological soliton solutions. The most famous and well-studied of these is theSkyrmion, which serves as a model of thenucleon that has withstood the test of time.

Consider a point particle, a charged particle, interacting with theelectromagnetic field. The interaction terms$${\displaystyle -q\varphi (\mathbf{x}(t),t)+q\dot{\mathbf{x}}(t)\cdot \mathbf{A}(\mathbf{x}(t),t)}$$are replaced by terms involving a continuous charge density ρ in A·s·m⁻³ and current density${\displaystyle \mathbf{j}}$ in A·m⁻². The resulting Lagrangian density for the electromagnetic field is:$${\displaystyle \mathcal{L}(\mathbf{x},t)=-\rho (\mathbf{x},t)\varphi (\mathbf{x},t)+\mathbf{j}(\mathbf{x},t)\cdot \mathbf{A}(\mathbf{x},t)+\frac{{ϵ}_0}{2}{E}^2(\mathbf{x},t)-\frac{1}{2{\mu }_0}{B}^2(\mathbf{x},t).}$$

Varying this with respect toϕ, we get$${\displaystyle 0=-\rho (\mathbf{x},t)+{ϵ}_0\mathrm{\nabla }\cdot \mathbf{E}(\mathbf{x},t)}$$which yieldsGauss' law.

Varying instead with respect to${\displaystyle \mathbf{A}}$, we get$${\displaystyle 0=\mathbf{j}(\mathbf{x},t)+{ϵ}_0\dot{\mathbf{E}}(\mathbf{x},t)-\frac{1}{{\mu }_0}\mathrm{\nabla }\times \mathbf{B}(\mathbf{x},t)}$$which yieldsAmpère's law.

Usingtensor notation, we can write all this more compactly. The term${\displaystyle -\rho \varphi (\mathbf{x},t)+\mathbf{j}\cdot \mathbf{A}}$ is actually the inner product of twofour-vectors. We package the charge density into the current 4-vector and the potential into the potential 4-vector. These two new vectors are$${\displaystyle j^{\mu }=(\rho ,\mathbf{j})\text{and}{A}_{\mu }=(-\varphi ,\mathbf{A})}$$We can then write the interaction term as$${\displaystyle -\rho \varphi +\mathbf{j}\cdot \mathbf{A}=j^{\mu }{A}_{\mu }}$$Additionally, we can package the E and B fields into what is known as theelectromagnetic tensor${\displaystyle F_{\mu \nu }}$.We define this tensor as$${\displaystyle F_{\mu \nu }={\mathrm{\partial }}_{\mu }{A}_{\nu }-{\mathrm{\partial }}_{\nu }{A}_{\mu }}$$The term we are looking out for turns out to be$${\displaystyle \frac{{ϵ}_0}{2}{E}^2-\frac{1}{2{\mu }_0}{B}^2=-\frac{1}{4{\mu }_0}{F}_{\mu \nu }{F}^{\mu \nu }=-\frac{1}{4{\mu }_0}{F}_{\mu \nu }{F}_{\rho \sigma }{\eta }^{\mu \rho }{\eta }^{\nu \sigma }}$$We have made use of theMinkowski metric to raise the indices on the EMF tensor. In this notation, Maxwell's equations are$${\displaystyle {\mathrm{\partial }}_{\mu }{F}^{\mu \nu }=-{\mu }_0{j}^{\nu }\text{and}{ϵ}^{\mu \nu \lambda \sigma }{\mathrm{\partial }}_{\nu }{F}_{\lambda \sigma }=0}$$where ε is theLevi-Civita tensor. So the Lagrange density for electromagnetism in special relativity written in terms of Lorentz vectors and tensors is$${\displaystyle \mathcal{L}(x)=j^{\mu }(x)A_{\mu }(x)-\frac{1}{4{\mu }_0}{F}_{\mu \nu }(x)F^{\mu \nu }(x)}$$In this notation it is apparent that classical electromagnetism is a Lorentz-invariant theory. By theequivalence principle, it becomes simple to extend the notion of electromagnetism to curved spacetime.^[7][8]

Usingdifferential forms, the electromagnetic actionS in vacuum on a (pseudo-) Riemannian manifold${\displaystyle \mathcal{M}}$ can be written (usingnatural units,c =ε₀ = 1) as$${\displaystyle \mathcal{S}[\mathbf{A}]=-{\int }_{\mathcal{M}}\left(\frac{1}{2}\mathbf{F}\wedge \ast \mathbf{F}-\mathbf{A}\wedge \ast \mathbf{J}\right).}$$Here,A stands for the electromagnetic potential 1-form,J is the current 1-form,F is the field strength 2-form and the star denotes theHodge star operator. This is exactly the same Lagrangian as in the section above, except that the treatment here is coordinate-free; expanding the integrand into a basis yields the identical, lengthy expression. Note that with forms, an additional integration measure is not necessary because forms have coordinate differentials built in. Variation of the action leads to$${\displaystyle \mathrm{d}\ast \mathbf{F}=\ast \mathbf{J}.}$$These are Maxwell's equations for the electromagnetic potential. SubstitutingF = dA immediately yields the equation for the fields,$${\displaystyle \mathrm{d}\mathbf{F}=0}$$becauseF is anexact form.

TheA field can be understood to be theaffine connection on aU(1)-fiber bundle. That is, classical electrodynamics, all of its effects and equations, can becompletely understood in terms of acircle bundle overMinkowski spacetime.

TheYang–Mills equations can be written in exactly the same form as above, by replacing theLie groupU(1) of electromagnetism by an arbitrary Lie group. In theStandard Model, it is conventionally taken to be${\displaystyle \mathrm{S}\mathrm{U}(3)\times \mathrm{S}\mathrm{U}(2)\times \mathrm{U}(1)}$ although the general case is of general interest. In all cases, there is no need for any quantization to be performed. Although the Yang–Mills equations are historically rooted in quantum field theory, the above equations are purely classical.^[2]

In the same vein as the above, one can consider the action in one dimension less, i.e. in acontact geometry setting. This gives theChern–Simons functional. It is written as$${\displaystyle \mathcal{S}[\mathbf{A}]={\int }_{\mathcal{M}}\mathrm{t}\mathrm{r}\left(\mathbf{A}\wedge d\mathbf{A}+\frac{2}{3}\mathbf{A}\wedge \mathbf{A}\wedge \mathbf{A}\right).}$$

Chern–Simons theory was deeply explored in physics, as a toy model for a broad range of geometric phenomena that might be expected to be found in aGrand Unified Theory.

The Lagrangian density forGinzburg–Landau theory combines the Lagrangian for thescalar field theory with the Lagrangian for theYang–Mills action. It may be written as:^[9]$${\displaystyle \mathcal{L}(\psi ,A)=|F{|}^2+|D\psi {|}^2+\frac{1}{4}{\left(\sigma -|\psi {|}^2\right)}^2}$$where${\displaystyle \psi }$ is asection of avector bundle with fiber${\displaystyle {\mathbb{C}}^n}$. The${\displaystyle \psi }$ corresponds to the order parameter in asuperconductor; equivalently, it corresponds to theHiggs field, after noting that the second term is the famous"Sombrero hat" potential. The field${\displaystyle A}$ is the (non-Abelian) gauge field, i.e. theYang–Mills field and${\displaystyle F}$ is its field-strength. TheEuler–Lagrange equations for the Ginzburg–Landau functional are theYang–Mills equations$${\displaystyle D\star D\psi =\frac{1}{2}\left(\sigma -|\psi {|}^2\right)\psi }$$and$${\displaystyle D\star F=-\mathrm{Re}⟨D\psi ,\psi ⟩}$$where${\displaystyle \star }$ is theHodge star operator, i.e. the fully antisymmetric tensor. These equations are closely related to theYang–Mills–Higgs equations. Another closely related Lagrangian is found inSeiberg–Witten theory.

The Lagrangian density for aDirac field is:^[10]: 143$${\displaystyle \mathcal{L}=\overline{\psi }(i\hslash c\mathrm{\partial }/-m{c}^2)\psi }$$where${\displaystyle \psi }$ is aDirac spinor,${\displaystyle \overline{\psi }={\psi }^{†}{\gamma }^0}$ is itsDirac adjoint, and${\displaystyle \mathrm{\partial }/}$ isFeynman slash notation for${\displaystyle {\gamma }^{\sigma }{\mathrm{\partial }}_{\sigma }}$.^{[dubious –discuss]} There is no particular need to focus on Dirac spinors in the classical theory. TheWeyl spinors provide a more general foundation; they can be constructed directly from theClifford algebra of spacetime; the construction works in any number of dimensions,^[11] and the Dirac spinors appear as a special case. Weyl spinors have the additional advantage that they can be used in avielbein for the metric on a Riemannian manifold; this enables the concept of aspin structure, which, roughly speaking, is a way of formulating spinors consistently in a curved spacetime.

The Lagrangian density forQED combines the Lagrangian for the Dirac field together with the Lagrangian for electrodynamics in a gauge-invariant way. It is:$${\displaystyle {\mathcal{L}}_{\mathrm{Q}\mathrm{E}\mathrm{D}}=\overline{\psi }(i\hslash cD/-m{c}^2)\psi -\frac{1}{4{\mu }_0}{F}_{\mu \nu }{F}^{\mu \nu }}$$where${\displaystyle F^{\mu \nu }}$ is theelectromagnetic tensor,D is thegauge covariant derivative, and${\displaystyle D/}$ isFeynman notation for${\displaystyle {\gamma }^{\sigma }{D}_{\sigma }}$ with${\displaystyle D_{\sigma }={\mathrm{\partial }}_{\sigma }-ie{A}_{\sigma }}$ where${\displaystyle A_{\sigma }}$ is theelectromagnetic four-potential. Although the word "quantum" appears in the above, this is a historical artifact. The definition of the Dirac field requires no quantization whatsoever, it can be written as a purely classical field of anti-commutingWeyl spinors constructed from first principles from aClifford algebra.^[11] The full gauge-invariant classical formulation is given in Bleecker.^[2]

The Lagrangian density forquantum chromodynamics combines the Lagrangian for one or more massiveDirac spinors with the Lagrangian for theYang–Mills action, which describes the dynamics of a gauge field; the combined Lagrangian is gauge invariant. It may be written as:^[12]$${\displaystyle {\mathcal{L}}_{\mathrm{Q}\mathrm{C}\mathrm{D}}=\sum _n{\overline{\psi }}_n\left(i\hslash cD/-m_n{c}^2\right){\psi }_n-\frac{1}{4}{G}^{\alpha }{}_{\mu \nu }{G}_{\alpha }{}^{\mu \nu }}$$whereD is the QCDgauge covariant derivative,n = 1, 2, ...6 counts thequark types, and${\displaystyle G^{\alpha }{}_{\mu \nu }}$ is thegluon field strength tensor. As for the electrodynamics case above, the appearance of the word "quantum" above only acknowledges its historical development. The Lagrangian and its gauge invariance can be formulated and treated in a purely classical fashion.^[2][11]

The Lagrange density for general relativity in the presence of matter fields is$${\displaystyle {\mathcal{L}}_{\text{GR}}={\mathcal{L}}_{\text{EH}}+{\mathcal{L}}_{\text{matter}}=\frac{c^4}{16\pi G}\left(R-2\mathrm{\Lambda }\right)+{\mathcal{L}}_{\text{matter}}}$$where${\displaystyle \mathrm{\Lambda }}$ is thecosmological constant,${\displaystyle R}$ is thecurvature scalar, which is theRicci tensor contracted with themetric tensor, and theRicci tensor is theRiemann tensor contracted with aKronecker delta. The integral of${\displaystyle {\mathcal{L}}_{\text{EH}}}$ is known as theEinstein–Hilbert action. The Riemann tensor is thetidal force tensor, and is constructed out ofChristoffel symbols and derivatives of Christoffel symbols, which define themetric connection on spacetime. The gravitational field itself was historically ascribed to the metric tensor; the modern view is that the connection is "more fundamental". This is due to the understanding that one can write connections with non-zerotorsion. These alter the metric without altering the geometry one bit. As to the actual "direction in which gravity points" (e.g. on the surface of the Earth, it points down), this comes from the Riemann tensor: it is the thing that describes the "gravitational force field" that moving bodies feel and react to. (This last statement must be qualified: there is no "force field"per se; moving bodies followgeodesics on the manifold described by the connection. They move in a "straight line".)

The Lagrangian for general relativity can also be written in a form that makes it manifestly similar to the Yang–Mills equations. This is called the Einstein–Yang–Mills action principle. This is done by noting that most of differential geometry works "just fine" on bundles with anaffine connection and arbitrary Lie group. Then, plugging in SO(3,1) for that symmetry group, i.e. for theframe fields, one obtains the equations above.^[2][11]

Substituting this Lagrangian into the Euler–Lagrange equation and taking the metric tensor${\displaystyle g_{\mu \nu }}$ as the field, we obtain theEinstein field equations$${\displaystyle R_{\mu \nu }-\frac{1}{2}R{g}_{\mu \nu }+g_{\mu \nu }\mathrm{\Lambda }=\frac{8\pi G}{c^4}{T}_{\mu \nu }.}$$${\displaystyle T_{\mu \nu }}$ is theenergy momentum tensor and is defined by$${\displaystyle T_{\mu \nu }\equiv \frac{-2}{\sqrt{-g}}\frac{\delta ({\mathcal{L}}_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}}\sqrt{-g})}{\delta g^{\mu \nu }}=-2\frac{\delta {\mathcal{L}}_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}}}{\delta g^{\mu \nu }}+g_{\mu \nu }{\mathcal{L}}_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}}.}$$where${\displaystyle g}$ is the determinant of the metric tensor when regarded as a matrix. Generally, in general relativity, the integration measure of the action of Lagrange density is$\sqrt{-g}{d}^{4}x$. This makes the integral coordinate independent, as the root of the metric determinant is equivalent to theJacobian determinant. The minus sign is a consequence of the metric signature (the determinant by itself is negative).^[7] This is an example of thevolume form, previously discussed, becoming manifest in non-flat spacetime.

The Lagrange density of electromagnetism in general relativity also contains the Einstein–Hilbert action from above. The pure electromagnetic Lagrangian is precisely a matter Lagrangian${\displaystyle {\mathcal{L}}_{\text{matter}}}$. The Lagrangian is

This Lagrangian is obtained by simply replacing the Minkowski metric in the above flat Lagrangian with a more general (possibly curved) metric${\displaystyle g_{\mu \nu }(x)}$. We can generate the Einstein Field Equations in the presence of an EM field using this lagrangian. The energy-momentum tensor is$${\displaystyle T^{\mu \nu }(x)=\frac{2}{\sqrt{-g(x)}}\frac{\delta }{\delta g_{\mu \nu }(x)}{\mathcal{S}}_{\text{Maxwell}}=\frac{1}{{\mu }_0}\left(F_{\lambda }^{\mu }(x)F^{\nu \lambda }(x)-\frac{1}{4}{g}^{\mu \nu }(x)F_{\rho \sigma }(x)F^{\rho \sigma }(x)\right)}$$It can be shown that this energy momentum tensor is traceless, i.e. that$${\displaystyle T=g_{\mu \nu }{T}^{\mu \nu }=0}$$If we take the trace of both sides of the Einstein Field Equations, we obtain$${\displaystyle R=-\frac{8\pi G}{c^4}T}$$So the tracelessness of the energy momentum tensor implies that the curvature scalar in an electromagnetic field vanishes. The Einstein equations are then$${\displaystyle R^{\mu \nu }=\frac{8\pi G}{c^4}\frac{1}{{\mu }_0}\left({F^{\mu }}_{\lambda }(x)F^{\nu \lambda }(x)-\frac{1}{4}{g}^{\mu \nu }(x)F_{\rho \sigma }(x)F^{\rho \sigma }(x)\right)}$$Additionally, Maxwell's equations are$${\displaystyle D_{\mu }{F}^{\mu \nu }=-{\mu }_0{j}^{\nu }}$$where${\displaystyle D_{\mu }}$ is thecovariant derivative. For free space, we can set the current tensor equal to zero,${\displaystyle j^{\mu }=0}$. Solving both Einstein and Maxwell's equations around a spherically symmetric mass distribution in free space leads to theReissner–Nordström charged black hole, with the defining line element (written innatural units and with chargeQ):^[7]$${\displaystyle \mathrm{d}{s}^2=\left(1-\frac{2M}{r}+\frac{Q^2}{r^2}\right)\mathrm{d}{t}^2-{\left(1-\frac{2M}{r}+\frac{Q^2}{r^2}\right)}^{-1}\mathrm{d}{r}^2-r^2\mathrm{d}{\mathrm{\Omega }}^2}$$

One possible way of unifying the electromagnetic and gravitational Lagrangians (by using a fifth dimension) is given byKaluza–Klein theory.^[2] Effectively, one constructs an affine bundle, just as for the Yang–Mills equations given earlier, and then considers the action separately on the 4-dimensional and the 1-dimensional parts. Suchfactorizations, such as the fact that the 7-sphere can be written as a product of the 4-sphere and the 3-sphere, or that the 11-sphere is a product of the 4-sphere and the 7-sphere, accounted for much of the early excitement that atheory of everything had been found. Unfortunately, the 7-sphere proved not large enough to enclose all of theStandard Model, dashing these hopes.

• TheBF model Lagrangian, short for "Background Field", describes a system with trivial dynamics, when written on a flat spacetime manifold. On a topologically non-trivial spacetime, the system will have non-trivial classical solutions, which may be interpreted assolitons orinstantons. A variety of extensions exist, forming the foundations fortopological field theories.

• Mathematical physics
• Classical field theory
• Calculus of variations
• Quantum field theory

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