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 ${\displaystyle \varphi (x,y,z,t)}$ so that the equations of motion are obtained by means of an action principle, written as:

$${\displaystyle {\frac {\delta {\mathcal {S}}}{\delta \varphi _{i}}}=0,}$$

${\displaystyle {\mathcal {S}}}$

${\displaystyle \varphi _{i}(s)}$

$${\displaystyle {\mathcal {S}}\left=\int {{\mathcal {L}}\left(\varphi _{i}(s),\left\{{\frac {\partial \varphi _{i}(s)}{\partial s^{\alpha }}}\right\},\{s^{\alpha }\}\right)\,\mathrm {d} ^{n}s},}$$

where the brackets denote ${\displaystyle \{\cdot ~\forall \alpha \}}$; and s = {s^α} 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, ${\displaystyle {\mathcal {L}}}$, is used to denote the density, and ${\displaystyle \mathrm {d} ^{n}s}$ 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. Abraham and Marsden's textbook^[1] provided the first comprehensive description of classical mechanics in terms of modern geometrical ideas, i.e., in terms of tangent manifolds, symplectic manifolds and contact geometry. Bleecker's textbook^[2] provided a comprehensive presentation of field theories in physics in terms of gauge invariant fiber bundles. Such formulations were known or suspected long before. Jost^[3] continues with a geometric presentation, clarifying the relation between Hamiltonian and Lagrangian forms, describing spin manifolds from first principles, etc. Current research focuses on non-rigid affine structures, (sometimes called "quantum structures") wherein one replaces occurrences of vector spaces by tensor algebras. This research is motivated by the breakthrough understanding of quantum groups as affine Lie algebras (Lie groups are, in a sense "rigid", as they are determined by their Lie algebra. When reformulated on a tensor algebra, they become "floppy", having infinite degrees of freedom; see e.g., Virasoro algebra.)

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 ${\displaystyle \varphi }$, the Lagrangian density will take the form:^{[nb 1][4]}

$${\displaystyle {\mathcal {L}}(\varphi ,{\boldsymbol {\nabla }}\varphi ,\partial \varphi /\partial t,\mathbf {x} ,t)}$$

For many scalar fields

$${\displaystyle {\mathcal {L}}(\varphi _{1},{\boldsymbol {\nabla }}\varphi _{1},\partial \varphi _{1}/\partial t,\ldots ,\varphi _{n},{\boldsymbol {\nabla }}\varphi _{n},\partial \varphi _{n}/\partial t,\ldots ,\mathbf {x} ,t)}$$

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 ${\displaystyle m}$ real-valued scalar fields, ${\displaystyle \varphi _{1},\dots ,\varphi _{m}}$, then the field manifold is ${\displaystyle \mathbb {R} ^{m}}$. If the field is a real vector field, then the field manifold is isomorphic to ${\displaystyle \mathbb {R} ^{n}}$.

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

$${\displaystyle {\mathcal {S}}=\int L\,\mathrm {d} t\,,}$$

${\displaystyle {\mathcal {L}}}$

$${\displaystyle {\mathcal {S}}=\int {\mathcal {L}}(\varphi ,{\boldsymbol {\nabla }}\varphi ,\partial \varphi /\partial t,\mathbf {x} ,t)\,\mathrm {d} ^{3}\mathbf {x} \,\mathrm {d} t.}$$

The spatial volume integral of the Lagrangian density is the Lagrangian; in 3D,