Lagrangian field theory is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom. One motivation for the development of the Lagrangian formalism on fields, and more generally, for classical field theory, is to provide a clear mathematical foundation for quantum 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 of partial differential equations. This enables the formulation of solutions on spaces with well-characterized properties, such as Sobolev spaces. It enables various theorems to be provided, ranging from proofs of existence to the uniform convergence of formal series to the general settings of potential theory. In addition, insight and clarity is obtained by generalizations to Riemannian manifolds and fiber 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 the Chern–Gauss–Bonnet theorem and the Riemann–Roch theorem to the Atiyah–Singer index theorem and Chern–Simons theory.
Overview In field theory, the independent variable is replaced by an event in spacetime (x, y, z, t), or more generally still by a point s on a Riemannian manifold. The dependent variables are replaced by the value of a field at that point in spacetime φ ( x , y , z , t ) {\displaystyle \varphi (x,y,z,t)} so that the equations of motion are obtained by means of an action principle, written as:
δ S δ φ i = 0 , {\displaystyle {\frac {\delta {\mathcal {S}}}{\delta \varphi _{i}}}=0,}
where the action, S {\displaystyle {\mathcal {S}}} , is a functional of the dependent variables φ i ( s ) {\displaystyle \varphi _{i}(s)} , their derivatives and s itself
S [ φ i ] = ∫ L ( φ i ( s ) , { ∂ φ i ( s ) ∂ s α } , { s α } ) d n s , {\displaystyle {\mathcal {S}}\left[\varphi _{i}\right]=\int {{\mathcal {L}}\left(\varphi _{i}(s),\left\{{\frac {\partial \varphi _{i}(s)}{\partial s^{\alpha }}}\right\},\{s^{\alpha }\}\right)\,\mathrm {d} ^{n}s},}
where the brackets denote { ⋅ ∀ α } {\displaystyle \{\cdot ~\forall \alpha \}} ; and s = {sα} denotes the set of n independent variables of the system, including the time variable, and is indexed by α = 1, 2, 3, ..., n. The calligraphic typeface, L {\displaystyle {\mathcal {L}}} , is used to denote the density, and d n s {\displaystyle \mathrm {d} ^{n}s} is the volume 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 a fiber bundle, wherein the Euler–Lagrange equations can be interpreted as specifying the geodesics on the fiber bundle, leading to topics like tangent manifolds, symplectic manifolds and contact geometry. The field theories of physics can be developed in terms of gauge invariant fiber bundles.
Definitions In Lagrangian field theory, the Lagrangian as a function of generalized 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 variable t is replaced by an event in spacetime (x, y, z, t) or still more generally by a point s on a manifold. Often, a "Lagrangian density" is simply referred to as a "Lagrangian".
Scalar fields For one scalar field φ {\displaystyle \varphi } , the Lagrangian density will take the form:
L ( φ , ∇ φ , ∂ φ / ∂ t , x , t ) {\displaystyle {\mathcal {L}}(\varphi ,{\boldsymbol {\nabla }}\varphi ,\partial \varphi /\partial t,\mathbf {x} ,t)}
For many scalar fields
L ( φ 1 , ∇ φ 1 , ∂ φ 1 / ∂ t , … , φ n , ∇ φ n , ∂ φ n / ∂ t , … , x , t ) {\displaystyle {\mathcal {L}}(\varphi _{1},{\boldsymbol {\nabla }}\varphi _{1},\partial \varphi _{1}/\partial t,\ldots ,\varphi _{n},{\boldsymbol {\nabla }}\varphi _{n},\partial \varphi _{n}/\partial t,\ldots ,\mathbf {x} ,t)}
In mathematical formulations, the scalar fields are understood to be coordinates on a fiber bundle, and the derivatives of the field are understood to be sections of the jet bundle.
Vector fields, tensor fields, spinor fields The above can be generalized for vector fields, tensor fields, and spinor 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 m {\displaystyle m} real-valued scalar fields, φ 1 , … , φ m {\displaystyle \varphi _{1},\dots ,\varphi _{m}} , then the field manifold is R m {\displaystyle \mathbb {R} ^{m}} . If the field is a real vector field, then the field manifold is isomorphic to R n {\displaystyle \mathbb {R} ^{n}} .
Action The time integral of the Lagrangian is called the action denoted by S. In field theory, a distinction is occasionally made between the Lagrangian L, of which the time integral is the action
S = ∫ L d t , {\displaystyle {\mathcal {S}}=\int L\,\mathrm {d} t\,,}
and the Lagrangian density L {\displaystyle {\mathcal {L}}} , which one integrates over all spacetime to get the action:
S [ φ ] = ∫ L ( φ , ∇ φ , ∂ φ / ∂ t , x , t ) d 3 x d t . {\displaystyle {\mathcal {S}}[\varphi ]=\int {\mathcal {L}}(\varphi ,{\boldsymbol {\nabla }}\varphi ,\partial \varphi /\partial t,\mathbf {x} ,t)\,\mathrm {d} ^{3}\mathbf {x} \,\mathrm {d} t.}
The spatial volume integral of the Lagrangian density is the Lagrangian; in 3D,
L = ∫ L d 3 x . {\displaystyle L=\int {\mathcal {L}}\,\mathrm {d} ^{3}\mathbf {x} \,.}
The action is often referred to as the "action functional", in that it is a function of the fields (and their derivatives).
Volume form In the presence of gravity or when using general curvilinear coordinates, the Lagrangian density L {\displaystyle {\mathcal {L}}} will include a factor of g {\textstyle {\sqrt {g}}} . This ensures that the action is invariant under general coordinate transformations. In mathematical literature, spacetime is taken to be a Riemannian manifold M {\displaystyle M} and the integral then becomes the volume form
S = ∫ M | g | d x 1 ∧ ⋯ ∧ d x m L {\displaystyle {\mathcal {S}}=\int _{M}{\sqrt {|g|}}dx^{1}\wedge \cdots \wedge dx^{m}{\mathcal {L}}}
Here, the ∧ {\displaystyle \wedge } is the wedge product and | g | {\textstyle {\sqrt {|g|}}} is the square root of the determinant | g | {\displaystyle |g|} of the metric tensor g {\displaystyle g} on M {\displaystyle M} . For flat spacetime (e.g., Minkowski spacetime), the unit volume is one, i.e. | g | = 1 {\textstyle {\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 − g {\textstyle {\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 ∗ ( 1 ) {\displaystyle *(1)} where ∗ {\displaystyle *} is the Hodge star. That is,
∗ ( 1 ) = | g | d x 1 ∧ ⋯ ∧ d x m {\displaystyle *(1)={\sqrt {|g|}}dx^{1}\wedge \cdots \wedge dx^{m}}
and so
S = ∫ M ∗ ( 1 ) L {\displaystyle {\mathcal {S}}=\int _{M}*(1){\mathcal {L}}}
Not infrequently, the notation above is considered to be entirely superfluous, and
S = ∫ M L {\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.
Euler–Lagrange equations The Euler–Lagrange equations describe the geodesic flow of the field φ {\displaystyle \varphi } as a function of time. Taking the variation with respect to φ {\displaystyle \varphi } , one obtains
0 = δ S δ φ = ∫ M ∗ ( 1 ) ( − ∂ μ ( ∂ L ∂ ( ∂ μ φ ) ) + ∂ L ∂ φ ) . {\displaystyle 0={\frac {\delta {\mathcal {S}}}{\delta \varphi }}=\int _{M}*(1)\left(-\partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\varphi )}}\right)+{\frac {\partial {\mathcal {L}}}{\partial \varphi }}\right).}
Solving, with respect to the boundary conditions, one obtains the Euler–Lagrange equations:
∂ L ∂ φ = ∂ μ ( ∂ L ∂ ( ∂ μ φ ) ) . {\displaystyle {\frac {\partial {\mathcal {L}}}{\partial \varphi }}=\partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\varphi )}}\right).}
Lagrangian terms Often the Lagrangian consists of a sum of polynomial terms, with the symmetries of the theory and the fields involved dictating the types of terms that are allowed. For example, in relativistic theories, each term must be Lorentz invariant while in a theory with a gauge field, they must be gauge invariant. Terms that contain the product of two fields and no derivatives are known as mass terms, with these giving mass to the fields. For example, a single real scalar field ϕ ( x ) {\displaystyle \phi (x)} of mass m {\displaystyle m} has a mass term given by
L m = − 1 2 m 2 ϕ 2 ( x ) . {\displaystyle {\mathcal {L}}_{m}=-{\frac {1}{2}}m^{2}\phi ^{2}(x).}
The other terms that have two fields, those with at least one derivative, are known as kinetic terms. They make fields dynamical, with most theories requiring a restriction of at most two derivatives in kinetic terms to preserve probabililties in a quantum theory. They are also usually positive-definite to ensure positive energies. For example, the kinetic term for a relativistic real scalar field is given by
L k = 1 2 ∂ μ ϕ ∂ μ ϕ . {\displaystyle {\mathcal {L}}_{k}={\frac {1}{2}}\partial _{\mu }\phi \partial ^{\mu }\phi .}
Fields with no kinetic terms can also be found, playing the role of auxiliary fields, background fields, or currents. Theories with only kinetic and mass terms, form free field theories. Any term with more than two fields per term is known as an interaction term. The presence of these gives rise to interacting theories where particles can scatter off each other. The coefficients in front of these terms are known as coupling constants and they dictate the strength of the interaction. For example, a quartic interaction in a real scalar field theory is given by
L i = − g 4 ! ϕ 4 , {\displaystyle {\mathcal {L}}_{i}=-{\frac {g}{4!}}\phi ^{4},}
where g {\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 a momentum dependence to the scattering term as can be seen by going into momentum space. Terms with only one field are known as tadpole terms since they give rise to tadpole Feynman diagrams. In theories with translational symmetries, such terms can usually be eliminated by redefining some of the fields though a shift. Constant terms, those with no fields, have no physical consequences in non-gravitational theories. 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 the partition function, so again play no role. Physically this is because in these theories there is no absolute energy scale as the potential energy can always be shifted by an arbitrary constant without altering the physics. However, in gravitational systems the constant terms are multiplied by the metric determinant, coupling them to the spacetime. They play the role of the cosmological constant, directly affecting the dynamics of the theory at both a classical and quantum level. Polynomial terms are often expressed with certain canonical normalizations, used to simplify the Feynman rules that are derived from them. Usually one divides by the product of the factorial of the multipicity of the fields. For example, in a theory with two real scalar fields, a term of the form g ϕ n φ m {\displaystyle g\phi ^{n}\varphi ^{m}} term would be divided by n ! m ! {\displaystyle n!m!} . Particles and antiparticles are distinguished in this counting, so that a complex scalar field term of the form g ′ ϕ ¯ p ϕ p {\displaystyle g'{\bar {\phi }}^{p}\phi ^{p}} is divided by p ! p ! {\displaystyle p!p!} rather than ( 2 p ) ! {\displaystyle (2p)!} .
Examples 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.
Newtonian gravity The Lagrangian density for Newtonian gravity is:
L ( x , t ) = − 1 8 π G ( ∇ Φ ( x , t ) ) 2 − ρ ( x , t ) Φ ( x , t ) {\displaystyle {\mathcal {L}}(\mathbf {x} ,t)=-{1 \over 8\pi G}(\nabla \Phi (\mathbf {x} ,t))^{2}-\rho (\mathbf {x} ,t)\Phi (\mathbf {x} ,t)}
where Φ is the gravitational potential, ρ is the mass density, and G in m3·kg−1·s−2 is the gravitational constant. The density L {\displaystyle {\mathcal {L}}} has units of J·m−3. Here the interaction term involves a continuous mass density ρ 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 L = T − V {\displaystyle {\mathcal {L}}=T-V} , with the T = − ( ∇ Φ ) 2 / 8 π G {\displaystyle T=-(\nabla \Phi )^{2}/8\pi G} providing a kinetic term, and the interaction V = ρ Φ {\displaystyle V=\rho \Phi } the potential term. See also Nordström'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 to Φ is:
δ L ( x , t ) = − ρ ( x , t ) δ Φ ( x , t ) − 2 8 π G ( ∇ Φ ( x , t ) ) ⋅ ( ∇ δ Φ ( x , t ) ) . {\displaystyle \delta {\mathcal {L}}(\mathbf {x} ,t)=-\rho (\mathbf {x} ,t)\delta \Phi (\mathbf {x} ,t)-{2 \over 8\pi G}(\nabla \Phi (\mathbf {x} ,t))\cdot (\nabla \delta \Phi (\mathbf {x} ,t)).}
After integrating by parts, discarding the total integral, and dividing out by δΦ the formula becomes:
0 = − ρ ( x , t ) + 1 4 π G ∇ ⋅ ∇ Φ ( x , t ) {\displaystyle 0=-\rho (\mathbf {x} ,t)+{\frac {1}{4\pi G}}\nabla \cdot \nabla \Phi (\mathbf {x} ,t)}
which is equivalent to:
4 π G ρ ( x , t ) = ∇ 2 Φ ( x , t ) {\displaystyle 4\pi G\rho (\mathbf {x} ,t)=\nabla ^{2}\Phi (\mathbf {x} ,t)}
which yields Gauss's law for gravity.
Scalar field theory
The Lagrangian for a scalar field moving in a potential V ( ϕ ) {\displaystyle V(\phi )} can be written as
L = 1 2 ∂ μ ϕ ∂ μ ϕ − V ( ϕ ) = 1 2 ∂ μ ϕ ∂ μ ϕ − 1 2 m 2 ϕ 2 − ∑ n = 3 ∞ 1 n ! g n ϕ n {\displaystyle {\mathcal {L}}={\frac {1}{2}}\partial ^{\mu }\phi \partial _{\mu }\phi -V(\phi )={\frac {1}{2}}\partial ^{\mu }\phi \partial _{\mu }\phi -{\frac {1}{2}}m^{2}\phi ^{2}-\sum _{n=3}^{\infty }{\frac {1}{n!}}g_{n}\phi ^{n}}
It is not at all an accident that the scalar theory resembles the undergraduate textbook Lagrangian L = T − V {\displaystyle L=T-V} for the kinetic term of a free point particle written as T = m v 2 / 2 {\displaystyle T=mv^{2}/2} . The scalar theory is the field-theory generalization of a particle moving in a potential. When the V ( ϕ ) {\displaystyle V(\phi )} is the Mexican hat potential, the resulting fields are termed the Higgs fields.
Sigma model Lagrangian
The sigma model describes the motion of a scalar point particle constrained to move on a Riemannian 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:
L = 1 2 d ϕ ∧ ∗ d ϕ {\displaystyle {\mathcal {L}}={\frac {1}{2}}\mathrm {d} \phi \wedge {*\mathrm {d} \phi }}
where the d {\displaystyle \mathrm {d} } is the differential. An equivalent expression is
L = 1 2 ∑ i = 1 n ∑ j = 1 n g i j ( ϕ ) ∂ μ ϕ i ∂ μ ϕ j {\displaystyle {\mathcal {L}}={\frac {1}{2}}\sum _{i=1}^{n}\sum _{j=1}^{n}g_{ij}(\phi )\;\partial ^{\mu }\phi _{i}\partial _{\mu }\phi _{j}}
with g i j {\displaystyle g_{ij}} the Riemannian metric on the manifold of the field; i.e. the fields ϕ i {\displaystyle \phi _{i}} are just local coordinates on the coordinate chart of the manifold. A third common form is
L = 1 2 t r ( L μ L μ ) {\displaystyle {\mathcal {L}}={\frac {1}{2}}\mathrm {tr} \left(L_{\mu }L^{\mu }\right)}
with
L μ = U − 1 ∂ μ U {\displaystyle L_{\mu }=U^{-1}\partial _{\mu }U}
and U ∈ S U ( N ) {\displaystyle U\in \mathrm {SU} (N)} , the Lie group SU(N). This group can be replaced by any Lie group, or, more generally, by a symmetric space. The trace is just the Killing 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 the Maurer–Cartan form to the base spacetime. In general, sigma models exhibit topological soliton solutions. The most famous and well-studied of these is the Skyrmion, which serves as a model of the nucleon that has withstood the test of time.
Electromagnetism in special relativity
Consider a point particle, a charged particle, interacting with the electromagnetic field. The interaction terms
− q ϕ ( x ( t ) , t ) + q x ˙ ( t ) ⋅ A ( x ( t ) , t ) {\displaystyle -q\phi (\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 j {\displaystyle \mathbf {j} } in A·m−2. The resulting Lagrangian density for the electromagnetic field is:
L ( x , t ) = − ρ ( x , t ) ϕ ( x , t ) + j ( x , t ) ⋅ A ( x , t ) + ϵ 0 2 E 2 ( x , t ) − 1 2 μ 0 B 2 ( x , t ) . {\displaystyle {\mathcal {L}}(\mathbf {x} ,t)=-\rho (\mathbf {x} ,t)\phi (\mathbf {x} ,t)+\mathbf {j} (\mathbf {x} ,t)\cdot \mathbf {A} (\mathbf {x} ,t)+{\epsilon _{0} \over 2}{E}^{2}(\mathbf {x} ,t)-{1 \over {2\mu _{0}}}{B}^{2}(\mathbf {x} ,t).}
Varying this with respect to ϕ, we get
0 = − ρ ( x , t ) + ϵ 0 ∇ ⋅ E ( x , t ) {\displaystyle 0=-\rho (\mathbf {x} ,t)+\epsilon _{0}\nabla \cdot \mathbf {E} (\mathbf {x} ,t)}
which yields Gauss' law. Varying instead with respect to A {\displaystyle \mathbf {A} } , we get
0 = j ( x , t ) + ϵ 0 E ˙ ( x , t ) − 1 μ 0 ∇ × B ( x , t ) {\displaystyle 0=\mathbf {j} (\mathbf {x} ,t)+\epsilon _{0}{\dot {\mathbf {E} }}(\mathbf {x} ,t)-{1 \over \mu _{0}}\nabla \times \mathbf {B} (\mathbf {x} ,t)}
which yields Ampère's law. Using tensor notation, we can write all this more compactly. The term − ρ ϕ ( x , t ) + j ⋅ A {\displaystyle -\rho \phi (\mathbf {x} ,t)+\mathbf {j} \cdot \mathbf {A} } is actually the inner product of two four-vectors. We package the charge density into the current 4-vector and the potential into the potential 4-vector. These two new vectors are
j μ = ( ρ , j ) and